:: 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 ) )
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 ) )
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]
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]
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]
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 )
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] ) ) )
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:]
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:]
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:]
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:] <> {} )
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 )
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 )
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
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]
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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 )]
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 )
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 = {}
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 )
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]}
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
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
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
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
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
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
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
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
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] )
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 ) )
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:]
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 )
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 )
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:]
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 ) )
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 ) )
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]
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]
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]
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 )
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:]
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:]
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:]
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:] <> {} )
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 )
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 )
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
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]
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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 )]
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 )
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 )
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]}
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
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
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
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
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
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
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
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
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
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] )
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 ) )
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:]
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 )
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 )
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:]
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;