:: MCART_4 semantic presentation
theorem Th1: :: MCART_4:1
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 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 Y holds
Y1 misses X ) )
theorem Th2: :: MCART_4:2
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 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 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 ;
func [c1,c2,c3,c4,c5,c6,c7] -> set equals :: MCART_4:def 1
[[x1,x2,x3,x4,x5,x6],x7];
correctness
coherence
[[x1,x2,x3,x4,x5,x6],x7] is set ;
;
end;
:: deftheorem Def1 defines [ MCART_4:def 1 :
for
x1,
x2,
x3,
x4,
x5,
x6,
x7 being
set holds
[x1,x2,x3,x4,x5,x6,x7] = [[x1,x2,x3,x4,x5,x6],x7];
theorem Th3: :: MCART_4:3
for
x1,
x2,
x3,
x4,
x5,
x6,
x7 being
set holds
[x1,x2,x3,x4,x5,x6,x7] = [[[[[[x1,x2],x3],x4],x5],x6],x7]
theorem Th4: :: MCART_4:4
canceled;
theorem Th5: :: MCART_4:5
for
x1,
x2,
x3,
x4,
x5,
x6,
x7 being
set holds
[x1,x2,x3,x4,x5,x6,x7] = [[x1,x2,x3,x4,x5],x6,x7]
theorem Th6: :: MCART_4:6
for
x1,
x2,
x3,
x4,
x5,
x6,
x7 being
set holds
[x1,x2,x3,x4,x5,x6,x7] = [[x1,x2,x3,x4],x5,x6,x7]
theorem Th7: :: MCART_4:7
for
x1,
x2,
x3,
x4,
x5,
x6,
x7 being
set holds
[x1,x2,x3,x4,x5,x6,x7] = [[x1,x2,x3],x4,x5,x6,x7]
theorem Th8: :: MCART_4:8
for
x1,
x2,
x3,
x4,
x5,
x6,
x7 being
set holds
[x1,x2,x3,x4,x5,x6,x7] = [[x1,x2],x3,x4,x5,x6,x7]
theorem Th9: :: MCART_4:9
for
x1,
x2,
x3,
x4,
x5,
x6,
x7,
y1,
y2,
y3,
y4,
y5,
y6,
y7 being
set st
[x1,x2,x3,x4,x5,x6,x7] = [y1,y2,y3,y4,y5,y6,y7] holds
(
x1 = y1 &
x2 = y2 &
x3 = y3 &
x4 = y4 &
x5 = y5 &
x6 = y6 &
x7 = y7 )
theorem Th10: :: MCART_4:10
for
X being
set st
X <> {} holds
ex
x being
set st
(
x in X & ( for
x1,
x2,
x3,
x4,
x5,
x6,
x7 being
set holds
( ( not
x1 in X & not
x2 in X ) or not
x = [x1,x2,x3,x4,x5,x6,x7] ) ) )
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 ;
func [:c1,c2,c3,c4,c5,c6,c7:] -> set equals :: MCART_4:def 2
[:[:X1,X2,X3,X4,X5,X6:],X7:];
correctness
coherence
[:[:X1,X2,X3,X4,X5,X6:],X7:] is set ;
;
end;
:: deftheorem Def2 defines [: MCART_4:def 2 :
for
X1,
X2,
X3,
X4,
X5,
X6,
X7 being
set holds
[:X1,X2,X3,X4,X5,X6,X7:] = [:[:X1,X2,X3,X4,X5,X6:],X7:];
theorem Th11: :: MCART_4:11
for
X1,
X2,
X3,
X4,
X5,
X6,
X7 being
set holds
[:X1,X2,X3,X4,X5,X6,X7:] = [:[:[:[:[:[:X1,X2:],X3:],X4:],X5:],X6:],X7:]
theorem Th12: :: MCART_4:12
canceled;
theorem Th13: :: MCART_4:13
for
X1,
X2,
X3,
X4,
X5,
X6,
X7 being
set holds
[:X1,X2,X3,X4,X5,X6,X7:] = [:[:X1,X2,X3,X4,X5:],X6,X7:]
theorem Th14: :: MCART_4:14
for
X1,
X2,
X3,
X4,
X5,
X6,
X7 being
set holds
[:X1,X2,X3,X4,X5,X6,X7:] = [:[:X1,X2,X3,X4:],X5,X6,X7:]
theorem Th15: :: MCART_4:15
for
X1,
X2,
X3,
X4,
X5,
X6,
X7 being
set holds
[:X1,X2,X3,X4,X5,X6,X7:] = [:[:X1,X2,X3:],X4,X5,X6,X7:]
theorem Th16: :: MCART_4:16
for
X1,
X2,
X3,
X4,
X5,
X6,
X7 being
set holds
[:X1,X2,X3,X4,X5,X6,X7:] = [:[:X1,X2:],X3,X4,X5,X6,X7:]
theorem Th17: :: MCART_4:17
for
X1,
X2,
X3,
X4,
X5,
X6,
X7 being
set holds
( (
X1 <> {} &
X2 <> {} &
X3 <> {} &
X4 <> {} &
X5 <> {} &
X6 <> {} &
X7 <> {} ) iff
[:X1,X2,X3,X4,X5,X6,X7:] <> {} )
theorem Th18: :: MCART_4:18
for
X1,
X2,
X3,
X4,
X5,
X6,
X7,
Y1,
Y2,
Y3,
Y4,
Y5,
Y6,
Y7 being
set st
X1 <> {} &
X2 <> {} &
X3 <> {} &
X4 <> {} &
X5 <> {} &
X6 <> {} &
X7 <> {} &
[:X1,X2,X3,X4,X5,X6,X7:] = [:Y1,Y2,Y3,Y4,Y5,Y6,Y7:] holds
(
X1 = Y1 &
X2 = Y2 &
X3 = Y3 &
X4 = Y4 &
X5 = Y5 &
X6 = Y6 &
X7 = Y7 )
theorem Th19: :: MCART_4:19
for
X1,
X2,
X3,
X4,
X5,
X6,
X7,
Y1,
Y2,
Y3,
Y4,
Y5,
Y6,
Y7 being
set st
[:X1,X2,X3,X4,X5,X6,X7:] <> {} &
[:X1,X2,X3,X4,X5,X6,X7:] = [:Y1,Y2,Y3,Y4,Y5,Y6,Y7:] holds
(
X1 = Y1 &
X2 = Y2 &
X3 = Y3 &
X4 = Y4 &
X5 = Y5 &
X6 = Y6 &
X7 = Y7 )
theorem Th20: :: MCART_4:20
for
X,
Y being
set st
[:X,X,X,X,X,X,X:] = [:Y,Y,Y,Y,Y,Y,Y:] holds
X = Y
theorem Th21: :: MCART_4:21
for
X1,
X2,
X3,
X4,
X5,
X6,
X7 being
set st
X1 <> {} &
X2 <> {} &
X3 <> {} &
X4 <> {} &
X5 <> {} &
X6 <> {} &
X7 <> {} holds
for
x being
Element of
[:X1,X2,X3,X4,X5,X6,X7:] 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 st
x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7]
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 ;
assume E59:
(
X1 <> {} &
X2 <> {} &
X3 <> {} &
X4 <> {} &
X5 <> {} &
X6 <> {} &
X7 <> {} )
;
let x be
Element of
[:X1,X2,X3,X4,X5,X6,X7:];
func c8 `1 -> Element of
a1 means :
Def3:
:: MCART_4:def 3
for
x1,
x2,
x3,
x4,
x5,
x6,
x7 being
set st
x = [x1,x2,x3,x4,x5,x6,x7] holds
it = x1;
existence
ex b1 being Element of X1 st
for x1, x2, x3, x4, x5, x6, x7 being set st x = [x1,x2,x3,x4,x5,x6,x7] holds
b1 = x1
uniqueness
for b1, b2 being Element of X1 st ( for x1, x2, x3, x4, x5, x6, x7 being set st x = [x1,x2,x3,x4,x5,x6,x7] holds
b1 = x1 ) & ( for x1, x2, x3, x4, x5, x6, x7 being set st x = [x1,x2,x3,x4,x5,x6,x7] holds
b2 = x1 ) holds
b1 = b2
func c8 `2 -> Element of
a2 means :
Def4:
:: MCART_4:def 4
for
x1,
x2,
x3,
x4,
x5,
x6,
x7 being
set st
x = [x1,x2,x3,x4,x5,x6,x7] holds
it = x2;
existence
ex b1 being Element of X2 st
for x1, x2, x3, x4, x5, x6, x7 being set st x = [x1,x2,x3,x4,x5,x6,x7] holds
b1 = x2
uniqueness
for b1, b2 being Element of X2 st ( for x1, x2, x3, x4, x5, x6, x7 being set st x = [x1,x2,x3,x4,x5,x6,x7] holds
b1 = x2 ) & ( for x1, x2, x3, x4, x5, x6, x7 being set st x = [x1,x2,x3,x4,x5,x6,x7] holds
b2 = x2 ) holds
b1 = b2
func c8 `3 -> Element of
a3 means :
Def5:
:: MCART_4:def 5
for
x1,
x2,
x3,
x4,
x5,
x6,
x7 being
set st
x = [x1,x2,x3,x4,x5,x6,x7] holds
it = x3;
existence
ex b1 being Element of X3 st
for x1, x2, x3, x4, x5, x6, x7 being set st x = [x1,x2,x3,x4,x5,x6,x7] holds
b1 = x3
uniqueness
for b1, b2 being Element of X3 st ( for x1, x2, x3, x4, x5, x6, x7 being set st x = [x1,x2,x3,x4,x5,x6,x7] holds
b1 = x3 ) & ( for x1, x2, x3, x4, x5, x6, x7 being set st x = [x1,x2,x3,x4,x5,x6,x7] holds
b2 = x3 ) holds
b1 = b2
func c8 `4 -> Element of
a4 means :
Def6:
:: MCART_4:def 6
for
x1,
x2,
x3,
x4,
x5,
x6,
x7 being
set st
x = [x1,x2,x3,x4,x5,x6,x7] holds
it = x4;
existence
ex b1 being Element of X4 st
for x1, x2, x3, x4, x5, x6, x7 being set st x = [x1,x2,x3,x4,x5,x6,x7] holds
b1 = x4
uniqueness
for b1, b2 being Element of X4 st ( for x1, x2, x3, x4, x5, x6, x7 being set st x = [x1,x2,x3,x4,x5,x6,x7] holds
b1 = x4 ) & ( for x1, x2, x3, x4, x5, x6, x7 being set st x = [x1,x2,x3,x4,x5,x6,x7] holds
b2 = x4 ) holds
b1 = b2
func c8 `5 -> Element of
a5 means :
Def7:
:: MCART_4:def 7
for
x1,
x2,
x3,
x4,
x5,
x6,
x7 being
set st
x = [x1,x2,x3,x4,x5,x6,x7] holds
it = x5;
existence
ex b1 being Element of X5 st
for x1, x2, x3, x4, x5, x6, x7 being set st x = [x1,x2,x3,x4,x5,x6,x7] holds
b1 = x5
uniqueness
for b1, b2 being Element of X5 st ( for x1, x2, x3, x4, x5, x6, x7 being set st x = [x1,x2,x3,x4,x5,x6,x7] holds
b1 = x5 ) & ( for x1, x2, x3, x4, x5, x6, x7 being set st x = [x1,x2,x3,x4,x5,x6,x7] holds
b2 = x5 ) holds
b1 = b2
func c8 `6 -> Element of
a6 means :
Def8:
:: MCART_4:def 8
for
x1,
x2,
x3,
x4,
x5,
x6,
x7 being
set st
x = [x1,x2,x3,x4,x5,x6,x7] holds
it = x6;
existence
ex b1 being Element of X6 st
for x1, x2, x3, x4, x5, x6, x7 being set st x = [x1,x2,x3,x4,x5,x6,x7] holds
b1 = x6
uniqueness
for b1, b2 being Element of X6 st ( for x1, x2, x3, x4, x5, x6, x7 being set st x = [x1,x2,x3,x4,x5,x6,x7] holds
b1 = x6 ) & ( for x1, x2, x3, x4, x5, x6, x7 being set st x = [x1,x2,x3,x4,x5,x6,x7] holds
b2 = x6 ) holds
b1 = b2
func c8 `7 -> Element of
a7 means :
Def9:
:: MCART_4:def 9
for
x1,
x2,
x3,
x4,
x5,
x6,
x7 being
set st
x = [x1,x2,x3,x4,x5,x6,x7] holds
it = x7;
existence
ex b1 being Element of X7 st
for x1, x2, x3, x4, x5, x6, x7 being set st x = [x1,x2,x3,x4,x5,x6,x7] holds
b1 = x7
uniqueness
for b1, b2 being Element of X7 st ( for x1, x2, x3, x4, x5, x6, x7 being set st x = [x1,x2,x3,x4,x5,x6,x7] holds
b1 = x7 ) & ( for x1, x2, x3, x4, x5, x6, x7 being set st x = [x1,x2,x3,x4,x5,x6,x7] holds
b2 = x7 ) holds
b1 = b2
end;
:: deftheorem Def3 defines `1 MCART_4:def 3 :
for
X1,
X2,
X3,
X4,
X5,
X6,
X7 being
set st
X1 <> {} &
X2 <> {} &
X3 <> {} &
X4 <> {} &
X5 <> {} &
X6 <> {} &
X7 <> {} holds
for
x being
Element of
[:X1,X2,X3,X4,X5,X6,X7:] for
b9 being
Element of
X1 holds
(
b9 = x `1 iff for
x1,
x2,
x3,
x4,
x5,
x6,
x7 being
set st
x = [x1,x2,x3,x4,x5,x6,x7] holds
b9 = x1 );
:: deftheorem Def4 defines `2 MCART_4:def 4 :
for
X1,
X2,
X3,
X4,
X5,
X6,
X7 being
set st
X1 <> {} &
X2 <> {} &
X3 <> {} &
X4 <> {} &
X5 <> {} &
X6 <> {} &
X7 <> {} holds
for
x being
Element of
[:X1,X2,X3,X4,X5,X6,X7:] for
b9 being
Element of
X2 holds
(
b9 = x `2 iff for
x1,
x2,
x3,
x4,
x5,
x6,
x7 being
set st
x = [x1,x2,x3,x4,x5,x6,x7] holds
b9 = x2 );
:: deftheorem Def5 defines `3 MCART_4:def 5 :
for
X1,
X2,
X3,
X4,
X5,
X6,
X7 being
set st
X1 <> {} &
X2 <> {} &
X3 <> {} &
X4 <> {} &
X5 <> {} &
X6 <> {} &
X7 <> {} holds
for
x being
Element of
[:X1,X2,X3,X4,X5,X6,X7:] for
b9 being
Element of
X3 holds
(
b9 = x `3 iff for
x1,
x2,
x3,
x4,
x5,
x6,
x7 being
set st
x = [x1,x2,x3,x4,x5,x6,x7] holds
b9 = x3 );
:: deftheorem Def6 defines `4 MCART_4:def 6 :
for
X1,
X2,
X3,
X4,
X5,
X6,
X7 being
set st
X1 <> {} &
X2 <> {} &
X3 <> {} &
X4 <> {} &
X5 <> {} &
X6 <> {} &
X7 <> {} holds
for
x being
Element of
[:X1,X2,X3,X4,X5,X6,X7:] for
b9 being
Element of
X4 holds
(
b9 = x `4 iff for
x1,
x2,
x3,
x4,
x5,
x6,
x7 being
set st
x = [x1,x2,x3,x4,x5,x6,x7] holds
b9 = x4 );
:: deftheorem Def7 defines `5 MCART_4:def 7 :
for
X1,
X2,
X3,
X4,
X5,
X6,
X7 being
set st
X1 <> {} &
X2 <> {} &
X3 <> {} &
X4 <> {} &
X5 <> {} &
X6 <> {} &
X7 <> {} holds
for
x being
Element of
[:X1,X2,X3,X4,X5,X6,X7:] for
b9 being
Element of
X5 holds
(
b9 = x `5 iff for
x1,
x2,
x3,
x4,
x5,
x6,
x7 being
set st
x = [x1,x2,x3,x4,x5,x6,x7] holds
b9 = x5 );
:: deftheorem Def8 defines `6 MCART_4:def 8 :
for
X1,
X2,
X3,
X4,
X5,
X6,
X7 being
set st
X1 <> {} &
X2 <> {} &
X3 <> {} &
X4 <> {} &
X5 <> {} &
X6 <> {} &
X7 <> {} holds
for
x being
Element of
[:X1,X2,X3,X4,X5,X6,X7:] for
b9 being
Element of
X6 holds
(
b9 = x `6 iff for
x1,
x2,
x3,
x4,
x5,
x6,
x7 being
set st
x = [x1,x2,x3,x4,x5,x6,x7] holds
b9 = x6 );
:: deftheorem Def9 defines `7 MCART_4:def 9 :
for
X1,
X2,
X3,
X4,
X5,
X6,
X7 being
set st
X1 <> {} &
X2 <> {} &
X3 <> {} &
X4 <> {} &
X5 <> {} &
X6 <> {} &
X7 <> {} holds
for
x being
Element of
[:X1,X2,X3,X4,X5,X6,X7:] for
b9 being
Element of
X7 holds
(
b9 = x `7 iff for
x1,
x2,
x3,
x4,
x5,
x6,
x7 being
set st
x = [x1,x2,x3,x4,x5,x6,x7] holds
b9 = x7 );
theorem Th22: :: MCART_4:22
for
X1,
X2,
X3,
X4,
X5,
X6,
X7 being
set st
X1 <> {} &
X2 <> {} &
X3 <> {} &
X4 <> {} &
X5 <> {} &
X6 <> {} &
X7 <> {} holds
for
x being
Element of
[:X1,X2,X3,X4,X5,X6,X7:] for
x1,
x2,
x3,
x4,
x5,
x6,
x7 being
set st
x = [x1,x2,x3,x4,x5,x6,x7] holds
(
x `1 = x1 &
x `2 = x2 &
x `3 = x3 &
x `4 = x4 &
x `5 = x5 &
x `6 = x6 &
x `7 = x7 )
by , , , , , , ;
theorem Th23: :: MCART_4:23
for
X1,
X2,
X3,
X4,
X5,
X6,
X7 being
set st
X1 <> {} &
X2 <> {} &
X3 <> {} &
X4 <> {} &
X5 <> {} &
X6 <> {} &
X7 <> {} holds
for
x being
Element of
[:X1,X2,X3,X4,X5,X6,X7:] holds
x = [(x `1 ),(x `2 ),(x `3 ),(x `4 ),(x `5 ),(x `6 ),(x `7 )]
theorem Th24: :: MCART_4:24
for
X1,
X2,
X3,
X4,
X5,
X6,
X7 being
set st
X1 <> {} &
X2 <> {} &
X3 <> {} &
X4 <> {} &
X5 <> {} &
X6 <> {} &
X7 <> {} holds
for
x being
Element of
[:X1,X2,X3,X4,X5,X6,X7:] holds
(
x `1 = (((((x `1 ) `1 ) `1 ) `1 ) `1 ) `1 &
x `2 = (((((x `1 ) `1 ) `1 ) `1 ) `1 ) `2 &
x `3 = ((((x `1 ) `1 ) `1 ) `1 ) `2 &
x `4 = (((x `1 ) `1 ) `1 ) `2 &
x `5 = ((x `1 ) `1 ) `2 &
x `6 = (x `1 ) `2 &
x `7 = x `2 )
theorem Th25: :: MCART_4:25
for
X1,
X2,
X3,
X4,
X5,
X6,
X7 being
set st (
X1 c= [:X1,X2,X3,X4,X5,X6,X7:] or
X1 c= [:X2,X3,X4,X5,X6,X7,X1:] or
X1 c= [:X3,X4,X5,X6,X7,X1,X2:] or
X1 c= [:X4,X5,X6,X7,X1,X2,X3:] or
X1 c= [:X5,X6,X7,X1,X2,X3,X4:] or
X1 c= [:X6,X7,X1,X2,X3,X4,X5:] or
X1 c= [:X7,X1,X2,X3,X4,X5,X6:] ) holds
X1 = {}
theorem Th26: :: MCART_4:26
for
X1,
X2,
X3,
X4,
X5,
X6,
X7,
Y1,
Y2,
Y3,
Y4,
Y5,
Y6,
Y7 being
set st
[:X1,X2,X3,X4,X5,X6,X7:] meets [:Y1,Y2,Y3,Y4,Y5,Y6,Y7:] holds
(
X1 meets Y1 &
X2 meets Y2 &
X3 meets Y3 &
X4 meets Y4 &
X5 meets Y5 &
X6 meets Y6 &
X7 meets Y7 )
theorem Th27: :: MCART_4:27
for
x1,
x2,
x3,
x4,
x5,
x6,
x7 being
set holds
[:{x1},{x2},{x3},{x4},{x5},{x6},{x7}:] = {[x1,x2,x3,x4,x5,x6,x7]}
theorem Th28: :: MCART_4:28
for
X1,
X2,
X3,
X4,
X5,
X6,
X7 being
set for
x being
Element of
[:X1,X2,X3,X4,X5,X6,X7:] st
X1 <> {} &
X2 <> {} &
X3 <> {} &
X4 <> {} &
X5 <> {} &
X6 <> {} &
X7 <> {} holds
for
x1,
x2,
x3,
x4,
x5,
x6,
x7 being
set st
x = [x1,x2,x3,x4,x5,x6,x7] holds
(
x `1 = x1 &
x `2 = x2 &
x `3 = x3 &
x `4 = x4 &
x `5 = x5 &
x `6 = x6 &
x `7 = x7 )
by , , , , , , ;
theorem Th29: :: MCART_4:29
for
y1,
X1,
X2,
X3,
X4,
X5,
X6,
X7 being
set for
x being
Element of
[:X1,X2,X3,X4,X5,X6,X7:] st
X1 <> {} &
X2 <> {} &
X3 <> {} &
X4 <> {} &
X5 <> {} &
X6 <> {} &
X7 <> {} & ( 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 st
x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] holds
y1 = xx1 ) holds
y1 = x `1
theorem Th30: :: MCART_4:30
for
y2,
X1,
X2,
X3,
X4,
X5,
X6,
X7 being
set for
x being
Element of
[:X1,X2,X3,X4,X5,X6,X7:] st
X1 <> {} &
X2 <> {} &
X3 <> {} &
X4 <> {} &
X5 <> {} &
X6 <> {} &
X7 <> {} & ( 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 st
x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] holds
y2 = xx2 ) holds
y2 = x `2
theorem Th31: :: MCART_4:31
for
y3,
X1,
X2,
X3,
X4,
X5,
X6,
X7 being
set for
x being
Element of
[:X1,X2,X3,X4,X5,X6,X7:] st
X1 <> {} &
X2 <> {} &
X3 <> {} &
X4 <> {} &
X5 <> {} &
X6 <> {} &
X7 <> {} & ( 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 st
x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] holds
y3 = xx3 ) holds
y3 = x `3
theorem Th32: :: MCART_4:32
for
y4,
X1,
X2,
X3,
X4,
X5,
X6,
X7 being
set for
x being
Element of
[:X1,X2,X3,X4,X5,X6,X7:] st
X1 <> {} &
X2 <> {} &
X3 <> {} &
X4 <> {} &
X5 <> {} &
X6 <> {} &
X7 <> {} & ( 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 st
x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] holds
y4 = xx4 ) holds
y4 = x `4
theorem Th33: :: MCART_4:33
for
y5,
X1,
X2,
X3,
X4,
X5,
X6,
X7 being
set for
x being
Element of
[:X1,X2,X3,X4,X5,X6,X7:] st
X1 <> {} &
X2 <> {} &
X3 <> {} &
X4 <> {} &
X5 <> {} &
X6 <> {} &
X7 <> {} & ( 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 st
x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] holds
y5 = xx5 ) holds
y5 = x `5
theorem Th34: :: MCART_4:34
for
y6,
X1,
X2,
X3,
X4,
X5,
X6,
X7 being
set for
x being
Element of
[:X1,X2,X3,X4,X5,X6,X7:] st
X1 <> {} &
X2 <> {} &
X3 <> {} &
X4 <> {} &
X5 <> {} &
X6 <> {} &
X7 <> {} & ( 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 st
x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] holds
y6 = xx6 ) holds
y6 = x `6
theorem Th35: :: MCART_4:35
for
y7,
X1,
X2,
X3,
X4,
X5,
X6,
X7 being
set for
x being
Element of
[:X1,X2,X3,X4,X5,X6,X7:] st
X1 <> {} &
X2 <> {} &
X3 <> {} &
X4 <> {} &
X5 <> {} &
X6 <> {} &
X7 <> {} & ( 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 st
x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] holds
y7 = xx7 ) holds
y7 = x `7
theorem Th36: :: MCART_4:36
for
y,
X1,
X2,
X3,
X4,
X5,
X6,
X7 being
set st
y in [:X1,X2,X3,X4,X5,X6,X7:] holds
ex
x1,
x2,
x3,
x4,
x5,
x6,
x7 being
set st
(
x1 in X1 &
x2 in X2 &
x3 in X3 &
x4 in X4 &
x5 in X5 &
x6 in X6 &
x7 in X7 &
y = [x1,x2,x3,x4,x5,x6,x7] )
theorem Th37: :: MCART_4:37
for
x1,
x2,
x3,
x4,
x5,
x6,
x7,
X1,
X2,
X3,
X4,
X5,
X6,
X7 being
set holds
(
[x1,x2,x3,x4,x5,x6,x7] in [:X1,X2,X3,X4,X5,X6,X7:] iff (
x1 in X1 &
x2 in X2 &
x3 in X3 &
x4 in X4 &
x5 in X5 &
x6 in X6 &
x7 in X7 ) )
theorem Th38: :: MCART_4:38
for
X1,
X2,
X3,
X4,
X5,
X6,
X7,
Z being
set st ( for
y being
set holds
(
y in Z iff ex
x1,
x2,
x3,
x4,
x5,
x6,
x7 being
set st
(
x1 in X1 &
x2 in X2 &
x3 in X3 &
x4 in X4 &
x5 in X5 &
x6 in X6 &
x7 in X7 &
y = [x1,x2,x3,x4,x5,x6,x7] ) ) ) holds
Z = [:X1,X2,X3,X4,X5,X6,X7:]
theorem Th39: :: MCART_4:39
for
X1,
X2,
X3,
X4,
X5,
X6,
X7,
Y1,
Y2,
Y3,
Y4,
Y5,
Y6,
Y7 being
set st
X1 <> {} &
X2 <> {} &
X3 <> {} &
X4 <> {} &
X5 <> {} &
X6 <> {} &
X7 <> {} &
Y1 <> {} &
Y2 <> {} &
Y3 <> {} &
Y4 <> {} &
Y5 <> {} &
Y6 <> {} &
Y7 <> {} holds
for
x being
Element of
[:X1,X2,X3,X4,X5,X6,X7:] for
y being
Element of
[:Y1,Y2,Y3,Y4,Y5,Y6,Y7:] 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 )
theorem Th40: :: MCART_4:40
for
X1,
X2,
X3,
X4,
X5,
X6,
X7 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
x being
Element of
[:X1,X2,X3,X4,X5,X6,X7:] st
x in [:A1,A2,A3,A4,A5,A6,A7:] 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 )
theorem Th41: :: MCART_4:41
for
X1,
X2,
X3,
X4,
X5,
X6,
X7,
Y1,
Y2,
Y3,
Y4,
Y5,
Y6,
Y7 being
set st
X1 c= Y1 &
X2 c= Y2 &
X3 c= Y3 &
X4 c= Y4 &
X5 c= Y5 &
X6 c= Y6 &
X7 c= Y7 holds
[:X1,X2,X3,X4,X5,X6,X7:] c= [:Y1,Y2,Y3,Y4,Y5,Y6,Y7:]
theorem Th42: :: MCART_4:42
for
X1,
X2,
X3,
X4,
X5,
X6,
X7 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 holds
[:A1,A2,A3,A4,A5,A6,A7:] is
Subset of
[:X1,X2,X3,X4,X5,X6,X7:] by ;