:: MCART_6 semantic presentation
theorem Th1: :: MCART_6:1
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,
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 Th2: :: MCART_6:2
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,
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 be
set ;
let x2 be
set ;
let x3 be
set ;
let x4 be
set ;
let x5 be
set ;
let x6 be
set ;
let x7 be
set ;
let x8 be
set ;
let x9 be
set ;
func [c1,c2,c3,c4,c5,c6,c7,c8,c9] -> set equals :: MCART_6:def 1
[[x1,x2,x3,x4,x5,x6,x7,x8],x9];
coherence
[[x1,x2,x3,x4,x5,x6,x7,x8],x9] is set
;
end;
:: deftheorem Def1 defines [ MCART_6:def 1 :
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 Th3: :: MCART_6:3
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 Th4: :: MCART_6:4
canceled;
theorem Th5: :: MCART_6:5
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 Th6: :: MCART_6:6
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 Th7: :: MCART_6:7
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 Th8: :: MCART_6:8
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 Th9: :: MCART_6:9
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 Th10: :: MCART_6:10
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 Th11: :: MCART_6:11
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 be
set ;
let X2 be
set ;
let X3 be
set ;
let X4 be
set ;
let X5 be
set ;
let X6 be
set ;
let X7 be
set ;
let X8 be
set ;
let X9 be
set ;
func [:c1,c2,c3,c4,c5,c6,c7,c8,c9:] -> set equals :: MCART_6:def 2
[:[:X1,X2,X3,X4,X5,X6,X7,X8:],X9:];
coherence
[:[:X1,X2,X3,X4,X5,X6,X7,X8:],X9:] is set
;
end;
:: deftheorem Def2 defines [: MCART_6:def 2 :
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 Th12: :: MCART_6:12
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 Th13: :: MCART_6:13
canceled;
theorem Th14: :: MCART_6:14
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 Th15: :: MCART_6:15
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 Th16: :: MCART_6:16
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 Th17: :: MCART_6:17
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 Th18: :: MCART_6:18
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 Th19: :: MCART_6:19
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 Th20: :: MCART_6:20
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 Th21: :: MCART_6:21
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 Th22: :: MCART_6:22
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 Th23: :: MCART_6:23
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 Th24: :: MCART_6:24
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 be
set ;
let X2 be
set ;
let X3 be
set ;
let X4 be
set ;
let X5 be
set ;
let X6 be
set ;
let X7 be
set ;
let X8 be
set ;
let X9 be
set ;
assume E73:
(
X1 <> {} &
X2 <> {} &
X3 <> {} &
X4 <> {} &
X5 <> {} &
X6 <> {} &
X7 <> {} &
X8 <> {} &
X9 <> {} )
;
let x be
Element of
[:X1,X2,X3,X4,X5,X6,X7,X8,X9:];
func c10 `1 -> Element of
a1 means :
Def3:
:: MCART_6:def 3
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 c10 `2 -> Element of
a2 means :
Def4:
:: MCART_6:def 4
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 c10 `3 -> Element of
a3 means :
Def5:
:: MCART_6:def 5
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 c10 `4 -> Element of
a4 means :
Def6:
:: MCART_6:def 6
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 c10 `5 -> Element of
a5 means :
Def7:
:: MCART_6:def 7
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 c10 `6 -> Element of
a6 means :
Def8:
:: MCART_6:def 8
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 c10 `7 -> Element of
a7 means :
Def9:
:: MCART_6:def 9
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 c10 `8 -> Element of
a8 means :
Def10:
:: MCART_6:def 10
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 c10 `9 -> Element of
a9 means :
Def11:
:: MCART_6:def 11
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 Def3 defines `1 MCART_6:def 3 :
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 Def4 defines `2 MCART_6:def 4 :
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 Def5 defines `3 MCART_6:def 5 :
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 Def6 defines `4 MCART_6:def 6 :
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 Def7 defines `5 MCART_6:def 7 :
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 Def8 defines `6 MCART_6:def 8 :
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 Def9 defines `7 MCART_6:def 9 :
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 Def10 defines `8 MCART_6:def 10 :
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 Def11 defines `9 MCART_6:def 11 :
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 Th25: :: MCART_6:25
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 , , , , , , , , ;
theorem Th26: :: MCART_6:26
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 Th27: :: MCART_6:27
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 Th28: :: MCART_6:28
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 Th29: :: MCART_6:29
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 Th30: :: MCART_6:30
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 , , , , , , , , ;
theorem Th31: :: MCART_6:31
for
X1,
X2,
X3,
X4,
X5,
X6,
X7,
X8,
X9,
y1 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 Th32: :: MCART_6:32
for
X1,
X2,
X3,
X4,
X5,
X6,
X7,
X8,
X9,
y2 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 Th33: :: MCART_6:33
for
X1,
X2,
X3,
X4,
X5,
X6,
X7,
X8,
X9,
y3 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 Th34: :: MCART_6:34
for
X1,
X2,
X3,
X4,
X5,
X6,
X7,
X8,
X9,
y4 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 Th35: :: MCART_6:35
for
X1,
X2,
X3,
X4,
X5,
X6,
X7,
X8,
X9,
y5 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 Th36: :: MCART_6:36
for
X1,
X2,
X3,
X4,
X5,
X6,
X7,
X8,
X9,
y6 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 Th37: :: MCART_6:37
for
X1,
X2,
X3,
X4,
X5,
X6,
X7,
X8,
X9,
y7 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 Th38: :: MCART_6:38
for
X1,
X2,
X3,
X4,
X5,
X6,
X7,
X8,
X9,
y8 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 Th39: :: MCART_6:39
for
X1,
X2,
X3,
X4,
X5,
X6,
X7,
X8,
X9,
y9 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 Th40: :: MCART_6:40
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 Th41: :: MCART_6:41
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 Th42: :: MCART_6:42
for
Z,
X1,
X2,
X3,
X4,
X5,
X6,
X7,
X8,
X9 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 Th43: :: MCART_6:43
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 Th44: :: MCART_6:44
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 Th45: :: MCART_6:45
for
X1,
Y1,
X2,
Y2,
X3,
Y3,
X4,
Y4,
X5,
Y5,
X6,
Y6,
X7,
Y7,
X8,
Y8,
X9,
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 Th46: :: MCART_6:46
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 ;