:: MCART_5 semantic presentation

theorem :: MCART_5:1
for X being set st X <> {} holds
ex Y being set st
( Y in X & ( for Y1, Y2, Y3, Y4, Y5, Y6, Y7, Y8, Y9, YA, YB, YC being set st Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in Y9 & Y9 in YA & YA in YB & YB in YC & YC in Y holds
Y1 misses X ) )
proof end;

theorem Th2: :: MCART_5:2
for X being set st X <> {} holds
ex Y being set st
( Y in X & ( for Y1, Y2, Y3, Y4, Y5, Y6, Y7, Y8, Y9, YA, YB, YC, YD being set st Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in Y9 & Y9 in YA & YA in YB & YB in YC & YC in YD & YD in Y holds
Y1 misses X ) )
proof end;

definition
let x1, x2, x3, x4, x5, x6, x7, x8 be set ;
func [x1,x2,x3,x4,x5,x6,x7,x8] -> set equals :: MCART_5:def 1
[[x1,x2,x3,x4,x5,x6,x7],x8];
correctness
coherence
[[x1,x2,x3,x4,x5,x6,x7],x8] is set
;
;
end;

:: deftheorem defines [ MCART_5:def 1 :
for x1, x2, x3, x4, x5, x6, x7, x8 being set holds [x1,x2,x3,x4,x5,x6,x7,x8] = [[x1,x2,x3,x4,x5,x6,x7],x8];

theorem Th3: :: MCART_5:3
for x1, x2, x3, x4, x5, x6, x7, x8 being set holds [x1,x2,x3,x4,x5,x6,x7,x8] = [[[[[[[x1,x2],x3],x4],x5],x6],x7],x8]
proof end;

theorem :: MCART_5:4
for x1, x2, x3, x4, x5, x6, x7, x8 being set holds [x1,x2,x3,x4,x5,x6,x7,x8] = [[x1,x2,x3,x4,x5,x6],x7,x8] by MCART_1:def 3;

theorem :: MCART_5:5
for x1, x2, x3, x4, x5, x6, x7, x8 being set holds [x1,x2,x3,x4,x5,x6,x7,x8] = [[x1,x2,x3,x4,x5],x6,x7,x8] by MCART_1:31;

theorem :: MCART_5:6
for x1, x2, x3, x4, x5, x6, x7, x8 being set holds [x1,x2,x3,x4,x5,x6,x7,x8] = [[x1,x2,x3,x4],x5,x6,x7,x8]
proof end;

theorem :: MCART_5:7
for x1, x2, x3, x4, x5, x6, x7, x8 being set holds [x1,x2,x3,x4,x5,x6,x7,x8] = [[x1,x2,x3],x4,x5,x6,x7,x8]
proof end;

theorem Th8: :: MCART_5:8
for x1, x2, x3, x4, x5, x6, x7, x8 being set holds [x1,x2,x3,x4,x5,x6,x7,x8] = [[x1,x2],x3,x4,x5,x6,x7,x8] by MCART_3:44;

theorem Th9: :: MCART_5:9
for x1, x2, x3, x4, x5, x6, x7, x8, y1, y2, y3, y4, y5, y6, y7, y8 being set st [x1,x2,x3,x4,x5,x6,x7,x8] = [y1,y2,y3,y4,y5,y6,y7,y8] holds
( x1 = y1 & x2 = y2 & x3 = y3 & x4 = y4 & x5 = y5 & x6 = y6 & x7 = y7 & x8 = y8 )
proof end;

theorem Th10: :: MCART_5:10
for X being set st X <> {} holds
ex y being set st
( y in X & ( for x1, x2, x3, x4, x5, x6, x7, x8 being set holds
( ( not x1 in X & not x2 in X ) or not y = [x1,x2,x3,x4,x5,x6,x7,x8] ) ) )
proof end;

definition
let X1, X2, X3, X4, X5, X6, X7, X8 be set ;
func [:X1,X2,X3,X4,X5,X6,X7,X8:] -> set equals :: MCART_5:def 2
[:[:X1,X2,X3,X4,X5,X6,X7:],X8:];
correctness
coherence
[:[:X1,X2,X3,X4,X5,X6,X7:],X8:] is set
;
;
end;

:: deftheorem defines [: MCART_5:def 2 :
for X1, X2, X3, X4, X5, X6, X7, X8 being set holds [:X1,X2,X3,X4,X5,X6,X7,X8:] = [:[:X1,X2,X3,X4,X5,X6,X7:],X8:];

theorem Th11: :: MCART_5:11
for X1, X2, X3, X4, X5, X6, X7, X8 being set holds [:X1,X2,X3,X4,X5,X6,X7,X8:] = [:[:[:[:[:[:[:X1,X2:],X3:],X4:],X5:],X6:],X7:],X8:]
proof end;

theorem :: MCART_5:12
for X1, X2, X3, X4, X5, X6, X7, X8 being set holds [:X1,X2,X3,X4,X5,X6,X7,X8:] = [:[:X1,X2,X3,X4,X5,X6:],X7,X8:] by ZFMISC_1:def 3;

theorem :: MCART_5:13
for X1, X2, X3, X4, X5, X6, X7, X8 being set holds [:X1,X2,X3,X4,X5,X6,X7,X8:] = [:[:X1,X2,X3,X4,X5:],X6,X7,X8:] by MCART_1:53;

theorem :: MCART_5:14
for X1, X2, X3, X4, X5, X6, X7, X8 being set holds [:X1,X2,X3,X4,X5,X6,X7,X8:] = [:[:X1,X2,X3,X4:],X5,X6,X7,X8:]
proof end;

theorem :: MCART_5:15
for X1, X2, X3, X4, X5, X6, X7, X8 being set holds [:X1,X2,X3,X4,X5,X6,X7,X8:] = [:[:X1,X2,X3:],X4,X5,X6,X7,X8:]
proof end;

theorem Th16: :: MCART_5:16
for X1, X2, X3, X4, X5, X6, X7, X8 being set holds [:X1,X2,X3,X4,X5,X6,X7,X8:] = [:[:X1,X2:],X3,X4,X5,X6,X7,X8:] by MCART_3:51;

theorem Th17: :: MCART_5:17
for X1, X2, X3, X4, X5, X6, X7, X8 being set holds
( ( X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} & X8 <> {} ) iff [:X1,X2,X3,X4,X5,X6,X7,X8:] <> {} )
proof end;

theorem Th18: :: MCART_5:18
for X1, X2, X3, X4, X5, X6, X7, X8, Y1, Y2, Y3, Y4, Y5, Y6, Y7, Y8 being set st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} & X8 <> {} & [:X1,X2,X3,X4,X5,X6,X7,X8:] = [:Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8:] holds
( X1 = Y1 & X2 = Y2 & X3 = Y3 & X4 = Y4 & X5 = Y5 & X6 = Y6 & X7 = Y7 & X8 = Y8 )
proof end;

theorem :: MCART_5:19
for X1, X2, X3, X4, X5, X6, X7, X8, Y1, Y2, Y3, Y4, Y5, Y6, Y7, Y8 being set st [:X1,X2,X3,X4,X5,X6,X7,X8:] <> {} & [:X1,X2,X3,X4,X5,X6,X7,X8:] = [:Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8:] holds
( X1 = Y1 & X2 = Y2 & X3 = Y3 & X4 = Y4 & X5 = Y5 & X6 = Y6 & X7 = Y7 & X8 = Y8 )
proof end;

theorem :: MCART_5:20
for X, Y being set st [:X,X,X,X,X,X,X,X:] = [:Y,Y,Y,Y,Y,Y,Y,Y:] holds
X = Y
proof end;

theorem Th21: :: MCART_5:21
for X1, X2, X3, X4, X5, X6, X7, X8 being set st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} & X8 <> {} holds
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8:] ex xx1 being Element of X1 ex xx2 being Element of X2 ex xx3 being Element of X3 ex xx4 being Element of X4 ex xx5 being Element of X5 ex xx6 being Element of X6 ex xx7 being Element of X7 ex xx8 being Element of X8 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8]
proof end;

definition
let X1, X2, X3, X4, X5, X6, X7, X8 be set ;
assume A1: ( X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} & X8 <> {} ) ;
let x be Element of [:X1,X2,X3,X4,X5,X6,X7,X8:];
func x `1 -> Element of X1 means :Def3: :: MCART_5:def 3
for x1, x2, x3, x4, x5, x6, x7, x8 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8] holds
it = x1;
existence
ex b1 being Element of X1 st
for x1, x2, x3, x4, x5, x6, x7, x8 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8] holds
b1 = x1
proof end;
uniqueness
for b1, b2 being Element of X1 st ( for x1, x2, x3, x4, x5, x6, x7, x8 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8] holds
b1 = x1 ) & ( for x1, x2, x3, x4, x5, x6, x7, x8 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8] holds
b2 = x1 ) holds
b1 = b2
proof end;
func x `2 -> Element of X2 means :Def4: :: MCART_5:def 4
for x1, x2, x3, x4, x5, x6, x7, x8 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8] holds
it = x2;
existence
ex b1 being Element of X2 st
for x1, x2, x3, x4, x5, x6, x7, x8 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8] holds
b1 = x2
proof end;
uniqueness
for b1, b2 being Element of X2 st ( for x1, x2, x3, x4, x5, x6, x7, x8 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8] holds
b1 = x2 ) & ( for x1, x2, x3, x4, x5, x6, x7, x8 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8] holds
b2 = x2 ) holds
b1 = b2
proof end;
func x `3 -> Element of X3 means :Def5: :: MCART_5:def 5
for x1, x2, x3, x4, x5, x6, x7, x8 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8] holds
it = x3;
existence
ex b1 being Element of X3 st
for x1, x2, x3, x4, x5, x6, x7, x8 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8] holds
b1 = x3
proof end;
uniqueness
for b1, b2 being Element of X3 st ( for x1, x2, x3, x4, x5, x6, x7, x8 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8] holds
b1 = x3 ) & ( for x1, x2, x3, x4, x5, x6, x7, x8 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8] holds
b2 = x3 ) holds
b1 = b2
proof end;
func x `4 -> Element of X4 means :Def6: :: MCART_5:def 6
for x1, x2, x3, x4, x5, x6, x7, x8 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8] holds
it = x4;
existence
ex b1 being Element of X4 st
for x1, x2, x3, x4, x5, x6, x7, x8 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8] holds
b1 = x4
proof end;
uniqueness
for b1, b2 being Element of X4 st ( for x1, x2, x3, x4, x5, x6, x7, x8 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8] holds
b1 = x4 ) & ( for x1, x2, x3, x4, x5, x6, x7, x8 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8] holds
b2 = x4 ) holds
b1 = b2
proof end;
func x `5 -> Element of X5 means :Def7: :: MCART_5:def 7
for x1, x2, x3, x4, x5, x6, x7, x8 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8] holds
it = x5;
existence
ex b1 being Element of X5 st
for x1, x2, x3, x4, x5, x6, x7, x8 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8] holds
b1 = x5
proof end;
uniqueness
for b1, b2 being Element of X5 st ( for x1, x2, x3, x4, x5, x6, x7, x8 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8] holds
b1 = x5 ) & ( for x1, x2, x3, x4, x5, x6, x7, x8 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8] holds
b2 = x5 ) holds
b1 = b2
proof end;
func x `6 -> Element of X6 means :Def8: :: MCART_5:def 8
for x1, x2, x3, x4, x5, x6, x7, x8 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8] holds
it = x6;
existence
ex b1 being Element of X6 st
for x1, x2, x3, x4, x5, x6, x7, x8 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8] holds
b1 = x6
proof end;
uniqueness
for b1, b2 being Element of X6 st ( for x1, x2, x3, x4, x5, x6, x7, x8 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8] holds
b1 = x6 ) & ( for x1, x2, x3, x4, x5, x6, x7, x8 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8] holds
b2 = x6 ) holds
b1 = b2
proof end;
func x `7 -> Element of X7 means :Def9: :: MCART_5:def 9
for x1, x2, x3, x4, x5, x6, x7, x8 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8] holds
it = x7;
existence
ex b1 being Element of X7 st
for x1, x2, x3, x4, x5, x6, x7, x8 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8] holds
b1 = x7
proof end;
uniqueness
for b1, b2 being Element of X7 st ( for x1, x2, x3, x4, x5, x6, x7, x8 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8] holds
b1 = x7 ) & ( for x1, x2, x3, x4, x5, x6, x7, x8 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8] holds
b2 = x7 ) holds
b1 = b2
proof end;
func x `8 -> Element of X8 means :Def10: :: MCART_5:def 10
for x1, x2, x3, x4, x5, x6, x7, x8 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8] holds
it = x8;
existence
ex b1 being Element of X8 st
for x1, x2, x3, x4, x5, x6, x7, x8 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8] holds
b1 = x8
proof end;
uniqueness
for b1, b2 being Element of X8 st ( for x1, x2, x3, x4, x5, x6, x7, x8 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8] holds
b1 = x8 ) & ( for x1, x2, x3, x4, x5, x6, x7, x8 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8] holds
b2 = x8 ) holds
b1 = b2
proof end;
end;

:: deftheorem Def3 defines `1 MCART_5:def 3 :
for X1, X2, X3, X4, X5, X6, X7, X8 being set st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} & X8 <> {} holds
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8:]
for b10 being Element of X1 holds
( b10 = x `1 iff for x1, x2, x3, x4, x5, x6, x7, x8 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8] holds
b10 = x1 );

:: deftheorem Def4 defines `2 MCART_5:def 4 :
for X1, X2, X3, X4, X5, X6, X7, X8 being set st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} & X8 <> {} holds
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8:]
for b10 being Element of X2 holds
( b10 = x `2 iff for x1, x2, x3, x4, x5, x6, x7, x8 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8] holds
b10 = x2 );

:: deftheorem Def5 defines `3 MCART_5:def 5 :
for X1, X2, X3, X4, X5, X6, X7, X8 being set st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} & X8 <> {} holds
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8:]
for b10 being Element of X3 holds
( b10 = x `3 iff for x1, x2, x3, x4, x5, x6, x7, x8 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8] holds
b10 = x3 );

:: deftheorem Def6 defines `4 MCART_5:def 6 :
for X1, X2, X3, X4, X5, X6, X7, X8 being set st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} & X8 <> {} holds
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8:]
for b10 being Element of X4 holds
( b10 = x `4 iff for x1, x2, x3, x4, x5, x6, x7, x8 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8] holds
b10 = x4 );

:: deftheorem Def7 defines `5 MCART_5:def 7 :
for X1, X2, X3, X4, X5, X6, X7, X8 being set st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} & X8 <> {} holds
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8:]
for b10 being Element of X5 holds
( b10 = x `5 iff for x1, x2, x3, x4, x5, x6, x7, x8 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8] holds
b10 = x5 );

:: deftheorem Def8 defines `6 MCART_5:def 8 :
for X1, X2, X3, X4, X5, X6, X7, X8 being set st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} & X8 <> {} holds
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8:]
for b10 being Element of X6 holds
( b10 = x `6 iff for x1, x2, x3, x4, x5, x6, x7, x8 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8] holds
b10 = x6 );

:: deftheorem Def9 defines `7 MCART_5:def 9 :
for X1, X2, X3, X4, X5, X6, X7, X8 being set st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} & X8 <> {} holds
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8:]
for b10 being Element of X7 holds
( b10 = x `7 iff for x1, x2, x3, x4, x5, x6, x7, x8 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8] holds
b10 = x7 );

:: deftheorem Def10 defines `8 MCART_5:def 10 :
for X1, X2, X3, X4, X5, X6, X7, X8 being set st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} & X8 <> {} holds
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8:]
for b10 being Element of X8 holds
( b10 = x `8 iff for x1, x2, x3, x4, x5, x6, x7, x8 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8] holds
b10 = x8 );

theorem :: MCART_5:22
for X1, X2, X3, X4, X5, X6, X7, X8 being set st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} & X8 <> {} holds
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8:]
for x1, x2, x3, x4, x5, x6, x7, x8 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8] holds
( x `1 = x1 & x `2 = x2 & x `3 = x3 & x `4 = x4 & x `5 = x5 & x `6 = x6 & x `7 = x7 & x `8 = x8 ) by Def3, Def4, Def5, Def6, Def7, Def8, Def9, Def10;

theorem Th23: :: MCART_5:23
for X1, X2, X3, X4, X5, X6, X7, X8 being set st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} & X8 <> {} holds
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8:] holds x = [(x `1 ),(x `2 ),(x `3 ),(x `4 ),(x `5 ),(x `6 ),(x `7 ),(x `8 )]
proof end;

theorem Th24: :: MCART_5:24
for X1, X2, X3, X4, X5, X6, X7, X8 being set st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} & X8 <> {} holds
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8:] holds
( x `1 = ((((((x `1 ) `1 ) `1 ) `1 ) `1 ) `1 ) `1 & x `2 = ((((((x `1 ) `1 ) `1 ) `1 ) `1 ) `1 ) `2 & x `3 = (((((x `1 ) `1 ) `1 ) `1 ) `1 ) `2 & x `4 = ((((x `1 ) `1 ) `1 ) `1 ) `2 & x `5 = (((x `1 ) `1 ) `1 ) `2 & x `6 = ((x `1 ) `1 ) `2 & x `7 = (x `1 ) `2 & x `8 = x `2 )
proof end;

theorem :: MCART_5:25
for X1, X2, X3, X4, X5, X6, X7, X8 being set st ( X1 c= [:X1,X2,X3,X4,X5,X6,X7,X8:] or X1 c= [:X2,X3,X4,X5,X6,X7,X8,X1:] or X1 c= [:X3,X4,X5,X6,X7,X8,X1,X2:] or X1 c= [:X4,X5,X6,X7,X8,X1,X2,X3:] or X1 c= [:X5,X6,X7,X8,X1,X2,X3,X4:] or X1 c= [:X6,X7,X8,X1,X2,X3,X4,X5:] or X1 c= [:X7,X8,X1,X2,X3,X4,X5,X6:] or X1 c= [:X8,X1,X2,X3,X4,X5,X6,X7:] ) holds
X1 = {}
proof end;

theorem Th26: :: MCART_5:26
for X1, X2, X3, X4, X5, X6, X7, X8, Y1, Y2, Y3, Y4, Y5, Y6, Y7, Y8 being set st [:X1,X2,X3,X4,X5,X6,X7,X8:] meets [:Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8:] holds
( X1 meets Y1 & X2 meets Y2 & X3 meets Y3 & X4 meets Y4 & X5 meets Y5 & X6 meets Y6 & X7 meets Y7 & X8 meets Y8 )
proof end;

theorem Th27: :: MCART_5:27
for x1, x2, x3, x4, x5, x6, x7, x8 being set holds [:{x1},{x2},{x3},{x4},{x5},{x6},{x7},{x8}:] = {[x1,x2,x3,x4,x5,x6,x7,x8]}
proof end;

theorem :: MCART_5:28
for X1, X2, X3, X4, X5, X6, X7, X8 being set
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8:] st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} & X8 <> {} holds
for x1, x2, x3, x4, x5, x6, x7, x8 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8] holds
( x `1 = x1 & x `2 = x2 & x `3 = x3 & x `4 = x4 & x `5 = x5 & x `6 = x6 & x `7 = x7 & x `8 = x8 ) by Def3, Def4, Def5, Def6, Def7, Def8, Def9, Def10;

theorem :: MCART_5:29
for y1, X1, X2, X3, X4, X5, X6, X7, X8 being set
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8:] st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} & X8 <> {} & ( for xx1 being Element of X1
for xx2 being Element of X2
for xx3 being Element of X3
for xx4 being Element of X4
for xx5 being Element of X5
for xx6 being Element of X6
for xx7 being Element of X7
for xx8 being Element of X8 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8] holds
y1 = xx1 ) holds
y1 = x `1
proof end;

theorem :: MCART_5:30
for y2, X1, X2, X3, X4, X5, X6, X7, X8 being set
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8:] st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} & X8 <> {} & ( for xx1 being Element of X1
for xx2 being Element of X2
for xx3 being Element of X3
for xx4 being Element of X4
for xx5 being Element of X5
for xx6 being Element of X6
for xx7 being Element of X7
for xx8 being Element of X8 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8] holds
y2 = xx2 ) holds
y2 = x `2
proof end;

theorem :: MCART_5:31
for y3, X1, X2, X3, X4, X5, X6, X7, X8 being set
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8:] st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} & X8 <> {} & ( for xx1 being Element of X1
for xx2 being Element of X2
for xx3 being Element of X3
for xx4 being Element of X4
for xx5 being Element of X5
for xx6 being Element of X6
for xx7 being Element of X7
for xx8 being Element of X8 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8] holds
y3 = xx3 ) holds
y3 = x `3
proof end;

theorem :: MCART_5:32
for y4, X1, X2, X3, X4, X5, X6, X7, X8 being set
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8:] st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} & X8 <> {} & ( for xx1 being Element of X1
for xx2 being Element of X2
for xx3 being Element of X3
for xx4 being Element of X4
for xx5 being Element of X5
for xx6 being Element of X6
for xx7 being Element of X7
for xx8 being Element of X8 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8] holds
y4 = xx4 ) holds
y4 = x `4
proof end;

theorem :: MCART_5:33
for y5, X1, X2, X3, X4, X5, X6, X7, X8 being set
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8:] st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} & X8 <> {} & ( for xx1 being Element of X1
for xx2 being Element of X2
for xx3 being Element of X3
for xx4 being Element of X4
for xx5 being Element of X5
for xx6 being Element of X6
for xx7 being Element of X7
for xx8 being Element of X8 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8] holds
y5 = xx5 ) holds
y5 = x `5
proof end;

theorem :: MCART_5:34
for y6, X1, X2, X3, X4, X5, X6, X7, X8 being set
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8:] st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} & X8 <> {} & ( for xx1 being Element of X1
for xx2 being Element of X2
for xx3 being Element of X3
for xx4 being Element of X4
for xx5 being Element of X5
for xx6 being Element of X6
for xx7 being Element of X7
for xx8 being Element of X8 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8] holds
y6 = xx6 ) holds
y6 = x `6
proof end;

theorem :: MCART_5:35
for y7, X1, X2, X3, X4, X5, X6, X7, X8 being set
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8:] st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} & X8 <> {} & ( for xx1 being Element of X1
for xx2 being Element of X2
for xx3 being Element of X3
for xx4 being Element of X4
for xx5 being Element of X5
for xx6 being Element of X6
for xx7 being Element of X7
for xx8 being Element of X8 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8] holds
y7 = xx7 ) holds
y7 = x `7
proof end;

theorem :: MCART_5:36
for y8, X1, X2, X3, X4, X5, X6, X7, X8 being set
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8:] st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} & X8 <> {} & ( for xx1 being Element of X1
for xx2 being Element of X2
for xx3 being Element of X3
for xx4 being Element of X4
for xx5 being Element of X5
for xx6 being Element of X6
for xx7 being Element of X7
for xx8 being Element of X8 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8] holds
y8 = xx8 ) holds
y8 = x `8
proof end;

theorem Th37: :: MCART_5:37
for y, X1, X2, X3, X4, X5, X6, X7, X8 being set st y in [:X1,X2,X3,X4,X5,X6,X7,X8:] holds
ex x1, x2, x3, x4, x5, x6, x7, x8 being set st
( x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5 & x6 in X6 & x7 in X7 & x8 in X8 & y = [x1,x2,x3,x4,x5,x6,x7,x8] )
proof end;

theorem Th38: :: MCART_5:38
for x1, x2, x3, x4, x5, x6, x7, x8, X1, X2, X3, X4, X5, X6, X7, X8 being set holds
( [x1,x2,x3,x4,x5,x6,x7,x8] in [:X1,X2,X3,X4,X5,X6,X7,X8:] iff ( x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5 & x6 in X6 & x7 in X7 & x8 in X8 ) )
proof end;

theorem :: MCART_5:39
for X1, X2, X3, X4, X5, X6, X7, X8, Z being set st ( for y being set holds
( y in Z iff ex x1, x2, x3, x4, x5, x6, x7, x8 being set st
( x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5 & x6 in X6 & x7 in X7 & x8 in X8 & y = [x1,x2,x3,x4,x5,x6,x7,x8] ) ) ) holds
Z = [:X1,X2,X3,X4,X5,X6,X7,X8:]
proof end;

theorem Th40: :: MCART_5:40
for X1, X2, X3, X4, X5, X6, X7, X8, Y1, Y2, Y3, Y4, Y5, Y6, Y7, Y8 being set st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} & X8 <> {} & Y1 <> {} & Y2 <> {} & Y3 <> {} & Y4 <> {} & Y5 <> {} & Y6 <> {} & Y7 <> {} & Y8 <> {} holds
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8:]
for y being Element of [:Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8:] st x = y holds
( x `1 = y `1 & x `2 = y `2 & x `3 = y `3 & x `4 = y `4 & x `5 = y `5 & x `6 = y `6 & x `7 = y `7 & x `8 = y `8 )
proof end;

theorem :: MCART_5:41
for X1, X2, X3, X4, X5, X6, X7, X8 being set
for A1 being Subset of X1
for A2 being Subset of X2
for A3 being Subset of X3
for A4 being Subset of X4
for A5 being Subset of X5
for A6 being Subset of X6
for A7 being Subset of X7
for A8 being Subset of X8
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8:] st x in [:A1,A2,A3,A4,A5,A6,A7,A8:] holds
( x `1 in A1 & x `2 in A2 & x `3 in A3 & x `4 in A4 & x `5 in A5 & x `6 in A6 & x `7 in A7 & x `8 in A8 )
proof end;

theorem Th42: :: MCART_5:42
for X1, X2, X3, X4, X5, X6, X7, X8, Y1, Y2, Y3, Y4, Y5, Y6, Y7, Y8 being set st X1 c= Y1 & X2 c= Y2 & X3 c= Y3 & X4 c= Y4 & X5 c= Y5 & X6 c= Y6 & X7 c= Y7 & X8 c= Y8 holds
[:X1,X2,X3,X4,X5,X6,X7,X8:] c= [:Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8:]
proof end;

theorem :: MCART_5:43
for X1, X2, X3, X4, X5, X6, X7, X8 being set
for A1 being Subset of X1
for A2 being Subset of X2
for A3 being Subset of X3
for A4 being Subset of X4
for A5 being Subset of X5
for A6 being Subset of X6
for A7 being Subset of X7
for A8 being Subset of X8 holds [:A1,A2,A3,A4,A5,A6,A7,A8:] is Subset of [:X1,X2,X3,X4,X5,X6,X7,X8:] by Th42;

theorem :: MCART_5:44
for X being set st X <> {} holds
ex Y being set st
( Y in X & ( for Y1, Y2, Y3, Y4, Y5, Y6, Y7, Y8, Y9, YA, YB, YC, YD, YE being set st Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in Y9 & Y9 in YA & YA in YB & YB in YC & YC in YD & YD in YE & YE in Y holds
Y1 misses X ) )
proof end;

theorem :: MCART_5:45
for X being set st X <> {} holds
ex Y being set st
( Y in X & ( for Y1, Y2, Y3, Y4, Y5, Y6, Y7, Y8, Y9, YA, YB, YC, YD, YE, YF being set st Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in Y9 & Y9 in YA & YA in YB & YB in YC & YC in YD & YD in YE & YE in YF & YF in Y holds
Y1 misses X ) )
proof end;

definition
let x1, x2, x3, x4, x5, x6, x7, x8, x9 be set ;
func [x1,x2,x3,x4,x5,x6,x7,x8,x9] -> set equals :: MCART_5:def 11
[[x1,x2,x3,x4,x5,x6,x7,x8],x9];
coherence
[[x1,x2,x3,x4,x5,x6,x7,x8],x9] is set
;
end;

:: deftheorem defines [ MCART_5:def 11 :
for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set holds [x1,x2,x3,x4,x5,x6,x7,x8,x9] = [[x1,x2,x3,x4,x5,x6,x7,x8],x9];

theorem Th46: :: MCART_5:46
for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set holds [x1,x2,x3,x4,x5,x6,x7,x8,x9] = [[[[[[[[x1,x2],x3],x4],x5],x6],x7],x8],x9]
proof end;

theorem :: MCART_5:47
for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set holds [x1,x2,x3,x4,x5,x6,x7,x8,x9] = [[x1,x2,x3,x4,x5,x6,x7],x8,x9] by MCART_1:def 3;

theorem :: MCART_5:48
for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set holds [x1,x2,x3,x4,x5,x6,x7,x8,x9] = [[x1,x2,x3,x4,x5,x6],x7,x8,x9] by MCART_1:31;

theorem :: MCART_5:49
for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set holds [x1,x2,x3,x4,x5,x6,x7,x8,x9] = [[x1,x2,x3,x4,x5],x6,x7,x8,x9] by MCART_2:3;

theorem :: MCART_5:50
for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set holds [x1,x2,x3,x4,x5,x6,x7,x8,x9] = [[x1,x2,x3,x4],x5,x6,x7,x8,x9]
proof end;

theorem :: MCART_5:51
for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set holds [x1,x2,x3,x4,x5,x6,x7,x8,x9] = [[x1,x2,x3],x4,x5,x6,x7,x8,x9]
proof end;

theorem Th52: :: MCART_5:52
for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set holds [x1,x2,x3,x4,x5,x6,x7,x8,x9] = [[x1,x2],x3,x4,x5,x6,x7,x8,x9] by Th8;

theorem Th53: :: MCART_5:53
for x1, x2, x3, x4, x5, x6, x7, x8, x9, y1, y2, y3, y4, y5, y6, y7, y8, y9 being set st [x1,x2,x3,x4,x5,x6,x7,x8,x9] = [y1,y2,y3,y4,y5,y6,y7,y8,y9] holds
( x1 = y1 & x2 = y2 & x3 = y3 & x4 = y4 & x5 = y5 & x6 = y6 & x7 = y7 & x8 = y8 & x9 = y9 )
proof end;

definition
let X1, X2, X3, X4, X5, X6, X7, X8, X9 be set ;
func [:X1,X2,X3,X4,X5,X6,X7,X8,X9:] -> set equals :: MCART_5:def 12
[:[:X1,X2,X3,X4,X5,X6,X7,X8:],X9:];
coherence
[:[:X1,X2,X3,X4,X5,X6,X7,X8:],X9:] is set
;
end;

:: deftheorem defines [: MCART_5:def 12 :
for X1, X2, X3, X4, X5, X6, X7, X8, X9 being set holds [:X1,X2,X3,X4,X5,X6,X7,X8,X9:] = [:[:X1,X2,X3,X4,X5,X6,X7,X8:],X9:];

theorem Th54: :: MCART_5:54
for X1, X2, X3, X4, X5, X6, X7, X8, X9 being set holds [:X1,X2,X3,X4,X5,X6,X7,X8,X9:] = [:[:[:[:[:[:[:[:X1,X2:],X3:],X4:],X5:],X6:],X7:],X8:],X9:]
proof end;

theorem :: MCART_5:55
for X1, X2, X3, X4, X5, X6, X7, X8, X9 being set holds [:X1,X2,X3,X4,X5,X6,X7,X8,X9:] = [:[:X1,X2,X3,X4,X5,X6,X7:],X8,X9:] by ZFMISC_1:def 3;

theorem :: MCART_5:56
for X1, X2, X3, X4, X5, X6, X7, X8, X9 being set holds [:X1,X2,X3,X4,X5,X6,X7,X8,X9:] = [:[:X1,X2,X3,X4,X5,X6:],X7,X8,X9:] by MCART_1:53;

theorem :: MCART_5:57
for X1, X2, X3, X4, X5, X6, X7, X8, X9 being set holds [:X1,X2,X3,X4,X5,X6,X7,X8,X9:] = [:[:X1,X2,X3,X4,X5:],X6,X7,X8,X9:] by MCART_2:9;

theorem :: MCART_5:58
for X1, X2, X3, X4, X5, X6, X7, X8, X9 being set holds [:X1,X2,X3,X4,X5,X6,X7,X8,X9:] = [:[:X1,X2,X3,X4:],X5,X6,X7,X8,X9:]
proof end;

theorem :: MCART_5:59
for X1, X2, X3, X4, X5, X6, X7, X8, X9 being set holds [:X1,X2,X3,X4,X5,X6,X7,X8,X9:] = [:[:X1,X2,X3:],X4,X5,X6,X7,X8,X9:]
proof end;

theorem Th60: :: MCART_5:60
for X1, X2, X3, X4, X5, X6, X7, X8, X9 being set holds [:X1,X2,X3,X4,X5,X6,X7,X8,X9:] = [:[:X1,X2:],X3,X4,X5,X6,X7,X8,X9:] by Th16;

theorem Th61: :: MCART_5:61
for X1, X2, X3, X4, X5, X6, X7, X8, X9 being set holds
( ( X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} & X8 <> {} & X9 <> {} ) iff [:X1,X2,X3,X4,X5,X6,X7,X8,X9:] <> {} )
proof end;

theorem Th62: :: MCART_5:62
for X1, X2, X3, X4, X5, X6, X7, X8, X9, Y1, Y2, Y3, Y4, Y5, Y6, Y7, Y8, Y9 being set st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} & X8 <> {} & X9 <> {} & [:X1,X2,X3,X4,X5,X6,X7,X8,X9:] = [:Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9:] holds
( X1 = Y1 & X2 = Y2 & X3 = Y3 & X4 = Y4 & X5 = Y5 & X6 = Y6 & X7 = Y7 & X8 = Y8 & X9 = Y9 )
proof end;

theorem :: MCART_5:63
for X1, X2, X3, X4, X5, X6, X7, X8, X9, Y1, Y2, Y3, Y4, Y5, Y6, Y7, Y8, Y9 being set st [:X1,X2,X3,X4,X5,X6,X7,X8,X9:] <> {} & [:X1,X2,X3,X4,X5,X6,X7,X8,X9:] = [:Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9:] holds
( X1 = Y1 & X2 = Y2 & X3 = Y3 & X4 = Y4 & X5 = Y5 & X6 = Y6 & X7 = Y7 & X8 = Y8 & X9 = Y9 )
proof end;

theorem :: MCART_5:64
for X, Y being set st [:X,X,X,X,X,X,X,X,X:] = [:Y,Y,Y,Y,Y,Y,Y,Y,Y:] holds
X = Y
proof end;

theorem Th65: :: MCART_5:65
for X1, X2, X3, X4, X5, X6, X7, X8, X9 being set st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} & X8 <> {} & X9 <> {} holds
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8,X9:] ex xx1 being Element of X1 ex xx2 being Element of X2 ex xx3 being Element of X3 ex xx4 being Element of X4 ex xx5 being Element of X5 ex xx6 being Element of X6 ex xx7 being Element of X7 ex xx8 being Element of X8 ex xx9 being Element of X9 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9]
proof end;

definition
let X1, X2, X3, X4, X5, X6, X7, X8, X9 be set ;
assume A1: ( X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} & X8 <> {} & X9 <> {} ) ;
let x be Element of [:X1,X2,X3,X4,X5,X6,X7,X8,X9:];
func x `1 -> Element of X1 means :Def13: :: MCART_5:def 13
for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] holds
it = x1;
existence
ex b1 being Element of X1 st
for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] holds
b1 = x1
proof end;
uniqueness
for b1, b2 being Element of X1 st ( for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] holds
b1 = x1 ) & ( for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] holds
b2 = x1 ) holds
b1 = b2
proof end;
func x `2 -> Element of X2 means :Def14: :: MCART_5:def 14
for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] holds
it = x2;
existence
ex b1 being Element of X2 st
for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] holds
b1 = x2
proof end;
uniqueness
for b1, b2 being Element of X2 st ( for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] holds
b1 = x2 ) & ( for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] holds
b2 = x2 ) holds
b1 = b2
proof end;
func x `3 -> Element of X3 means :Def15: :: MCART_5:def 15
for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] holds
it = x3;
existence
ex b1 being Element of X3 st
for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] holds
b1 = x3
proof end;
uniqueness
for b1, b2 being Element of X3 st ( for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] holds
b1 = x3 ) & ( for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] holds
b2 = x3 ) holds
b1 = b2
proof end;
func x `4 -> Element of X4 means :Def16: :: MCART_5:def 16
for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] holds
it = x4;
existence
ex b1 being Element of X4 st
for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] holds
b1 = x4
proof end;
uniqueness
for b1, b2 being Element of X4 st ( for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] holds
b1 = x4 ) & ( for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] holds
b2 = x4 ) holds
b1 = b2
proof end;
func x `5 -> Element of X5 means :Def17: :: MCART_5:def 17
for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] holds
it = x5;
existence
ex b1 being Element of X5 st
for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] holds
b1 = x5
proof end;
uniqueness
for b1, b2 being Element of X5 st ( for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] holds
b1 = x5 ) & ( for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] holds
b2 = x5 ) holds
b1 = b2
proof end;
func x `6 -> Element of X6 means :Def18: :: MCART_5:def 18
for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] holds
it = x6;
existence
ex b1 being Element of X6 st
for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] holds
b1 = x6
proof end;
uniqueness
for b1, b2 being Element of X6 st ( for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] holds
b1 = x6 ) & ( for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] holds
b2 = x6 ) holds
b1 = b2
proof end;
func x `7 -> Element of X7 means :Def19: :: MCART_5:def 19
for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] holds
it = x7;
existence
ex b1 being Element of X7 st
for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] holds
b1 = x7
proof end;
uniqueness
for b1, b2 being Element of X7 st ( for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] holds
b1 = x7 ) & ( for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] holds
b2 = x7 ) holds
b1 = b2
proof end;
func x `8 -> Element of X8 means :Def20: :: MCART_5:def 20
for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] holds
it = x8;
existence
ex b1 being Element of X8 st
for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] holds
b1 = x8
proof end;
uniqueness
for b1, b2 being Element of X8 st ( for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] holds
b1 = x8 ) & ( for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] holds
b2 = x8 ) holds
b1 = b2
proof end;
func x `9 -> Element of X9 means :Def21: :: MCART_5:def 21
for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] holds
it = x9;
existence
ex b1 being Element of X9 st
for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] holds
b1 = x9
proof end;
uniqueness
for b1, b2 being Element of X9 st ( for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] holds
b1 = x9 ) & ( for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] holds
b2 = x9 ) holds
b1 = b2
proof end;
end;

:: deftheorem Def13 defines `1 MCART_5:def 13 :
for X1, X2, X3, X4, X5, X6, X7, X8, X9 being set st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} & X8 <> {} & X9 <> {} holds
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8,X9:]
for b11 being Element of X1 holds
( b11 = x `1 iff for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] holds
b11 = x1 );

:: deftheorem Def14 defines `2 MCART_5:def 14 :
for X1, X2, X3, X4, X5, X6, X7, X8, X9 being set st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} & X8 <> {} & X9 <> {} holds
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8,X9:]
for b11 being Element of X2 holds
( b11 = x `2 iff for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] holds
b11 = x2 );

:: deftheorem Def15 defines `3 MCART_5:def 15 :
for X1, X2, X3, X4, X5, X6, X7, X8, X9 being set st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} & X8 <> {} & X9 <> {} holds
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8,X9:]
for b11 being Element of X3 holds
( b11 = x `3 iff for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] holds
b11 = x3 );

:: deftheorem Def16 defines `4 MCART_5:def 16 :
for X1, X2, X3, X4, X5, X6, X7, X8, X9 being set st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} & X8 <> {} & X9 <> {} holds
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8,X9:]
for b11 being Element of X4 holds
( b11 = x `4 iff for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] holds
b11 = x4 );

:: deftheorem Def17 defines `5 MCART_5:def 17 :
for X1, X2, X3, X4, X5, X6, X7, X8, X9 being set st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} & X8 <> {} & X9 <> {} holds
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8,X9:]
for b11 being Element of X5 holds
( b11 = x `5 iff for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] holds
b11 = x5 );

:: deftheorem Def18 defines `6 MCART_5:def 18 :
for X1, X2, X3, X4, X5, X6, X7, X8, X9 being set st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} & X8 <> {} & X9 <> {} holds
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8,X9:]
for b11 being Element of X6 holds
( b11 = x `6 iff for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] holds
b11 = x6 );

:: deftheorem Def19 defines `7 MCART_5:def 19 :
for X1, X2, X3, X4, X5, X6, X7, X8, X9 being set st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} & X8 <> {} & X9 <> {} holds
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8,X9:]
for b11 being Element of X7 holds
( b11 = x `7 iff for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] holds
b11 = x7 );

:: deftheorem Def20 defines `8 MCART_5:def 20 :
for X1, X2, X3, X4, X5, X6, X7, X8, X9 being set st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} & X8 <> {} & X9 <> {} holds
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8,X9:]
for b11 being Element of X8 holds
( b11 = x `8 iff for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] holds
b11 = x8 );

:: deftheorem Def21 defines `9 MCART_5:def 21 :
for X1, X2, X3, X4, X5, X6, X7, X8, X9 being set st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} & X8 <> {} & X9 <> {} holds
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8,X9:]
for b11 being Element of X9 holds
( b11 = x `9 iff for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] holds
b11 = x9 );

theorem :: MCART_5:66
for X1, X2, X3, X4, X5, X6, X7, X8, X9 being set st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} & X8 <> {} & X9 <> {} holds
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8,X9:]
for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] holds
( x `1 = x1 & x `2 = x2 & x `3 = x3 & x `4 = x4 & x `5 = x5 & x `6 = x6 & x `7 = x7 & x `8 = x8 & x `9 = x9 ) by Def13, Def14, Def15, Def16, Def17, Def18, Def19, Def20, Def21;

theorem Th67: :: MCART_5:67
for X1, X2, X3, X4, X5, X6, X7, X8, X9 being set st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} & X8 <> {} & X9 <> {} holds
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8,X9:] holds x = [(x `1 ),(x `2 ),(x `3 ),(x `4 ),(x `5 ),(x `6 ),(x `7 ),(x `8 ),(x `9 )]
proof end;

theorem Th68: :: MCART_5:68
for X1, X2, X3, X4, X5, X6, X7, X8, X9 being set st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} & X8 <> {} & X9 <> {} holds
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8,X9:] holds
( x `1 = (((((((x `1 ) `1 ) `1 ) `1 ) `1 ) `1 ) `1 ) `1 & x `2 = (((((((x `1 ) `1 ) `1 ) `1 ) `1 ) `1 ) `1 ) `2 & x `3 = ((((((x `1 ) `1 ) `1 ) `1 ) `1 ) `1 ) `2 & x `4 = (((((x `1 ) `1 ) `1 ) `1 ) `1 ) `2 & x `5 = ((((x `1 ) `1 ) `1 ) `1 ) `2 & x `6 = (((x `1 ) `1 ) `1 ) `2 & x `7 = ((x `1 ) `1 ) `2 & x `8 = (x `1 ) `2 & x `9 = x `2 )
proof end;

theorem :: MCART_5:69
for X1, X2, X3, X4, X5, X6, X7, X8, X9, Y1, Y2, Y3, Y4, Y5, Y6, Y7, Y8, Y9 being set st [:X1,X2,X3,X4,X5,X6,X7,X8,X9:] meets [:Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9:] holds
( X1 meets Y1 & X2 meets Y2 & X3 meets Y3 & X4 meets Y4 & X5 meets Y5 & X6 meets Y6 & X7 meets Y7 & X8 meets Y8 & X9 meets Y9 )
proof end;

theorem :: MCART_5:70
for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set holds [:{x1},{x2},{x3},{x4},{x5},{x6},{x7},{x8},{x9}:] = {[x1,x2,x3,x4,x5,x6,x7,x8,x9]}
proof end;

theorem :: MCART_5:71
for X1, X2, X3, X4, X5, X6, X7, X8, X9 being set
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8,X9:] st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} & X8 <> {} & X9 <> {} holds
for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] holds
( x `1 = x1 & x `2 = x2 & x `3 = x3 & x `4 = x4 & x `5 = x5 & x `6 = x6 & x `7 = x7 & x `8 = x8 & x `9 = x9 ) by Def13, Def14, Def15, Def16, Def17, Def18, Def19, Def20, Def21;

theorem :: MCART_5:72
for y1, X1, X2, X3, X4, X5, X6, X7, X8, X9 being set
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8,X9:] st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} & X8 <> {} & X9 <> {} & ( for xx1 being Element of X1
for xx2 being Element of X2
for xx3 being Element of X3
for xx4 being Element of X4
for xx5 being Element of X5
for xx6 being Element of X6
for xx7 being Element of X7
for xx8 being Element of X8
for xx9 being Element of X9 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9] holds
y1 = xx1 ) holds
y1 = x `1
proof end;

theorem :: MCART_5:73
for y2, X1, X2, X3, X4, X5, X6, X7, X8, X9 being set
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8,X9:] st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} & X8 <> {} & X9 <> {} & ( for xx1 being Element of X1
for xx2 being Element of X2
for xx3 being Element of X3
for xx4 being Element of X4
for xx5 being Element of X5
for xx6 being Element of X6
for xx7 being Element of X7
for xx8 being Element of X8
for xx9 being Element of X9 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9] holds
y2 = xx2 ) holds
y2 = x `2
proof end;

theorem :: MCART_5:74
for y3, X1, X2, X3, X4, X5, X6, X7, X8, X9 being set
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8,X9:] st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} & X8 <> {} & X9 <> {} & ( for xx1 being Element of X1
for xx2 being Element of X2
for xx3 being Element of X3
for xx4 being Element of X4
for xx5 being Element of X5
for xx6 being Element of X6
for xx7 being Element of X7
for xx8 being Element of X8
for xx9 being Element of X9 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9] holds
y3 = xx3 ) holds
y3 = x `3
proof end;

theorem :: MCART_5:75
for y4, X1, X2, X3, X4, X5, X6, X7, X8, X9 being set
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8,X9:] st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} & X8 <> {} & X9 <> {} & ( for xx1 being Element of X1
for xx2 being Element of X2
for xx3 being Element of X3
for xx4 being Element of X4
for xx5 being Element of X5
for xx6 being Element of X6
for xx7 being Element of X7
for xx8 being Element of X8
for xx9 being Element of X9 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9] holds
y4 = xx4 ) holds
y4 = x `4
proof end;

theorem :: MCART_5:76
for y5, X1, X2, X3, X4, X5, X6, X7, X8, X9 being set
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8,X9:] st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} & X8 <> {} & X9 <> {} & ( for xx1 being Element of X1
for xx2 being Element of X2
for xx3 being Element of X3
for xx4 being Element of X4
for xx5 being Element of X5
for xx6 being Element of X6
for xx7 being Element of X7
for xx8 being Element of X8
for xx9 being Element of X9 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9] holds
y5 = xx5 ) holds
y5 = x `5
proof end;

theorem :: MCART_5:77
for y6, X1, X2, X3, X4, X5, X6, X7, X8, X9 being set
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8,X9:] st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} & X8 <> {} & X9 <> {} & ( for xx1 being Element of X1
for xx2 being Element of X2
for xx3 being Element of X3
for xx4 being Element of X4
for xx5 being Element of X5
for xx6 being Element of X6
for xx7 being Element of X7
for xx8 being Element of X8
for xx9 being Element of X9 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9] holds
y6 = xx6 ) holds
y6 = x `6
proof end;

theorem :: MCART_5:78
for y7, X1, X2, X3, X4, X5, X6, X7, X8, X9 being set
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8,X9:] st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} & X8 <> {} & X9 <> {} & ( for xx1 being Element of X1
for xx2 being Element of X2
for xx3 being Element of X3
for xx4 being Element of X4
for xx5 being Element of X5
for xx6 being Element of X6
for xx7 being Element of X7
for xx8 being Element of X8
for xx9 being Element of X9 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9] holds
y7 = xx7 ) holds
y7 = x `7
proof end;

theorem :: MCART_5:79
for y8, X1, X2, X3, X4, X5, X6, X7, X8, X9 being set
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8,X9:] st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} & X8 <> {} & X9 <> {} & ( for xx1 being Element of X1
for xx2 being Element of X2
for xx3 being Element of X3
for xx4 being Element of X4
for xx5 being Element of X5
for xx6 being Element of X6
for xx7 being Element of X7
for xx8 being Element of X8
for xx9 being Element of X9 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9] holds
y8 = xx8 ) holds
y8 = x `8
proof end;

theorem :: MCART_5:80
for y9, X1, X2, X3, X4, X5, X6, X7, X8, X9 being set
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8,X9:] st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} & X8 <> {} & X9 <> {} & ( for xx1 being Element of X1
for xx2 being Element of X2
for xx3 being Element of X3
for xx4 being Element of X4
for xx5 being Element of X5
for xx6 being Element of X6
for xx7 being Element of X7
for xx8 being Element of X8
for xx9 being Element of X9 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9] holds
y9 = xx9 ) holds
y9 = x `9
proof end;

theorem :: MCART_5:81
for y, X1, X2, X3, X4, X5, X6, X7, X8, X9 being set st y in [:X1,X2,X3,X4,X5,X6,X7,X8,X9:] holds
ex x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st
( x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5 & x6 in X6 & x7 in X7 & x8 in X8 & x9 in X9 & y = [x1,x2,x3,x4,x5,x6,x7,x8,x9] )
proof end;

theorem :: MCART_5:82
for x1, x2, x3, x4, x5, x6, x7, x8, x9, X1, X2, X3, X4, X5, X6, X7, X8, X9 being set holds
( [x1,x2,x3,x4,x5,x6,x7,x8,x9] in [:X1,X2,X3,X4,X5,X6,X7,X8,X9:] iff ( x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5 & x6 in X6 & x7 in X7 & x8 in X8 & x9 in X9 ) )
proof end;

theorem :: MCART_5:83
for X1, X2, X3, X4, X5, X6, X7, X8, X9, Z being set st ( for y being set holds
( y in Z iff ex x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st
( x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5 & x6 in X6 & x7 in X7 & x8 in X8 & x9 in X9 & y = [x1,x2,x3,x4,x5,x6,x7,x8,x9] ) ) ) holds
Z = [:X1,X2,X3,X4,X5,X6,X7,X8,X9:]
proof end;

theorem Th84: :: MCART_5:84
for X1, X2, X3, X4, X5, X6, X7, X8, X9, Y1, Y2, Y3, Y4, Y5, Y6, Y7, Y8, Y9 being set st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {} & X8 <> {} & X9 <> {} & Y1 <> {} & Y2 <> {} & Y3 <> {} & Y4 <> {} & Y5 <> {} & Y6 <> {} & Y7 <> {} & Y8 <> {} & Y9 <> {} holds
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8,X9:]
for y being Element of [:Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9:] st x = y holds
( x `1 = y `1 & x `2 = y `2 & x `3 = y `3 & x `4 = y `4 & x `5 = y `5 & x `6 = y `6 & x `7 = y `7 & x `8 = y `8 & x `9 = y `9 )
proof end;

theorem :: MCART_5:85
for X1, X2, X3, X4, X5, X6, X7, X8, X9 being set
for A1 being Subset of X1
for A2 being Subset of X2
for A3 being Subset of X3
for A4 being Subset of X4
for A5 being Subset of X5
for A6 being Subset of X6
for A7 being Subset of X7
for A8 being Subset of X8
for A9 being Subset of X9
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8,X9:] st x in [:A1,A2,A3,A4,A5,A6,A7,A8,A9:] holds
( x `1 in A1 & x `2 in A2 & x `3 in A3 & x `4 in A4 & x `5 in A5 & x `6 in A6 & x `7 in A7 & x `8 in A8 & x `9 in A9 )
proof end;

theorem Th86: :: MCART_5:86
for X1, X2, X3, X4, X5, X6, X7, X8, X9, Y1, Y2, Y3, Y4, Y5, Y6, Y7, Y8, Y9 being set st X1 c= Y1 & X2 c= Y2 & X3 c= Y3 & X4 c= Y4 & X5 c= Y5 & X6 c= Y6 & X7 c= Y7 & X8 c= Y8 & X9 c= Y9 holds
[:X1,X2,X3,X4,X5,X6,X7,X8,X9:] c= [:Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9:]
proof end;

theorem :: MCART_5:87
for X1, X2, X3, X4, X5, X6, X7, X8, X9 being set
for A1 being Subset of X1
for A2 being Subset of X2
for A3 being Subset of X3
for A4 being Subset of X4
for A5 being Subset of X5
for A6 being Subset of X6
for A7 being Subset of X7
for A8 being Subset of X8
for A9 being Subset of X9 holds [:A1,A2,A3,A4,A5,A6,A7,A8,A9:] is Subset of [:X1,X2,X3,X4,X5,X6,X7,X8,X9:] by Th86;