### \$N\$-Tuples and Cartesian Products for \$n=7\$

by
Michal Muzalewski, and
Wojciech Skaba

Copyright (c) 1990 Association of Mizar Users

MML identifier: MCART_4
[ MML identifier index ]

```environ

vocabulary BOOLE, TARSKI, MCART_1, MCART_2, MCART_3, MCART_4;
notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, MCART_1, MCART_2, MCART_3;
constructors TARSKI, MCART_1, MCART_2, MCART_3, MEMBERED, XBOOLE_0;
clusters MEMBERED, ZFMISC_1, XBOOLE_0;
requirements SUBSET, BOOLE;
theorems TARSKI, ZFMISC_1, MCART_1, MCART_2, MCART_3, XBOOLE_0, XBOOLE_1;
schemes XBOOLE_0;

begin

reserve x,x1,x2,x3,x4,x5,x6,x7 for set;
reserve y,y1,y2,y3,y4,y5,y6,y7 for set;
reserve X,X1,X2,X3,X4,X5,X6,X7 for set;
reserve Y,Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9,YA,YB for set;
reserve Z,Z1,Z2,Z3,Z4,Z5,Z6,Z7,Z8,Z9,ZA,ZB for set;
reserve xx1 for Element of X1;
reserve xx2 for Element of X2;
reserve xx3 for Element of X3;
reserve xx4 for Element of X4;
reserve xx5 for Element of X5;
reserve xx6 for Element of X6;

theorem
X <> {} implies
ex Y st Y in X &
for Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9,YA
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
proof
defpred P1[set] means
ex Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9
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 \$1 & Y1 meets X;
consider Z1 such that
A1: for Y holds Y in Z1 iff Y in union X & P1[Y] from Separation;
defpred P2[set] means
ex Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8
st Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7
& Y7 in Y8 & Y8 in \$1 & Y1 meets X;
consider Z2 such that
A2:  for Y holds Y in Z2 iff Y in union union X & P2[Y] from Separation;
defpred P3[set] means
ex Y1,Y2,Y3,Y4,Y5,Y6,Y7
st Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7
& Y7 in \$1 & Y1 meets X;
consider Z3 such that
A3:  for Y holds Y in Z3 iff Y in union union union X & P3[Y] from Separation;
defpred P4[set] means
ex Y1,Y2,Y3,Y4,Y5,Y6
st Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 &
Y6 in \$1 & Y1 meets X;
consider Z4 such that
A4:  for Y holds Y in Z4 iff
Y in union union union union X & P4[Y] from Separation;
defpred P5[set] means
ex Y1,Y2,Y3,Y4,Y5
st Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in \$1 & Y1 meets X;
consider Z5 such that
A5:  for Y holds Y in Z5 iff
Y in union union union union union X & P5[Y] from Separation;
defpred P5[set] means
ex Y1,Y2,Y3,Y4 st Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in \$1 &  Y1 meets X;
consider Z6 such that
A6:  for Y holds Y in Z6 iff
Y in union union union union union union X & P5[Y] from Separation;
defpred P6[set] means
ex Y1,Y2,Y3 st Y1 in Y2 & Y2 in Y3 & Y3 in \$1 & Y1 meets X;
consider Z7 such that
A7:  for Y holds Y in Z7 iff
Y in union union union union union union union X & P6[Y]
from Separation;
defpred P7[set] means ex Y1,Y2 st Y1 in Y2 & Y2 in \$1 & Y1 meets X;
consider Z8 such that
A8:  for Y holds Y in Z8 iff
Y in union union union union union union union union X & P7[Y]
from Separation;
defpred P8[set] means ex Y1 st Y1 in \$1 & Y1 meets X;
consider Z9 such that
A9:  for Y holds Y in Z9 iff
Y in union union union union union union union union union X & P8[Y]
from Separation;
defpred P9[set] means \$1 meets X;
consider ZA such that
A10:  for Y holds Y in ZA iff
Y in union union union union union union union union union union X
& P9[Y] from Separation;

set V = X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA;

A11: V = X \/ (Z1 \/ Z2) \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3) \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4) \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5) \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6) \/ Z7 \/ Z8 \/ Z9 \/
ZA by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7) \/ Z8 \/ Z9 \/
ZA by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8) \/ Z9 \/
ZA by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9) \/
ZA by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA) by XBOOLE_1:4;

assume X <> {};
then V <> {} by A11,XBOOLE_1:15;
then consider Y such that
A12:  Y in V and
A13:  Y misses V by MCART_1:1;
assume
A14: not thesis;
now assume
A15:  Y in X;
then consider Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9,YA such that
A16:   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 & not Y1 misses X by A14;

YA in union X & Y1 meets X by A15,A16,TARSKI:def 4;
then YA in Z1 by A1,A16;
then YA in X \/ Z1 by XBOOLE_0:def 2;
then YA in X \/ Z1 \/ Z2 by XBOOLE_0:def 2;
then YA in X \/ Z1 \/ Z2 \/ Z3 by XBOOLE_0:def 2;
then YA in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_0:def 2;
then YA in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_0:def 2;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by A16,XBOOLE_0:3;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/
Z8 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/
Z9 by XBOOLE_1:70;
end;
then Y in Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA by A11,A12,XBOOLE_0:def 2;
then Y in Z1 \/ (Z2 \/ Z3) \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA by XBOOLE_1:4;
then Y in Z1 \/ (Z2 \/ Z3 \/ Z4) \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA by XBOOLE_1:4;
then Y in Z1 \/ (Z2 \/ Z3 \/ Z4 \/ Z5) \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA by XBOOLE_1:4;
then Y in Z1 \/ (Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6) \/ Z7 \/ Z8 \/ Z9 \/
ZA by XBOOLE_1:4;
then Y in Z1 \/ (Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7) \/ Z8 \/ Z9 \/
ZA by XBOOLE_1:4;
then Y in Z1 \/ (Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8) \/ Z9 \/
ZA by XBOOLE_1:4;
then Y in Z1 \/ (Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9) \/
ZA by XBOOLE_1:4;
then A17: Y in Z1 \/ (Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA) by XBOOLE_1:4;
now assume
A18:  Y in Z1;
then consider Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9 such that
A19:    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 Y & Y1 meets X by A1;
Y in union X by A1,A18;
then Y9 in union union X by A19,TARSKI:def 4;
then Y9 in Z2 by A2,A19;
then Y9 in X \/ Z1 \/ Z2 by XBOOLE_0:def 2;
then Y meets X \/ Z1 \/ Z2 by A19,XBOOLE_0:3;
then Y meets X \/ Z1 \/ Z2 \/ Z3 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/
Z8 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/
Z9 by XBOOLE_1:70;
end;
then Y in Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA by A17,XBOOLE_0:def 2;
then Y in Z2 \/ (Z3 \/ Z4) \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA by XBOOLE_1:4
;
then Y in Z2 \/ (Z3 \/ Z4 \/ Z5) \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA by XBOOLE_1:4
;
then Y in Z2 \/ (Z3 \/ Z4 \/ Z5 \/ Z6) \/ Z7 \/ Z8 \/ Z9 \/ ZA by XBOOLE_1:4
;
then Y in Z2 \/ (Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7) \/ Z8 \/ Z9 \/ ZA by XBOOLE_1:4
;
then Y in Z2 \/ (Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8) \/ Z9 \/ ZA by XBOOLE_1:4
;
then Y in Z2 \/ (Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9) \/
ZA by XBOOLE_1:4;
then A20: Y in Z2 \/ (Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA) by XBOOLE_1
:4;
now assume
A21:  Y in Z2;
then consider Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8 such that
A22:    Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in
Y8
& Y8 in Y & Y1 meets X by A2;
Y in union union X by A2,A21;
then Y8 in union union union X by A22,TARSKI:def 4;
then Y8 in Z3 by A3,A22;
then Y8 in X \/ Z1 \/ Z2 \/ Z3 by XBOOLE_0:def 2;
then Y8 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_0:def 2;
then Y8 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_0:def 2;
then Y8 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_0:def 2;
then Y8 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_0:def 2;
then Y8 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/
Z8 by XBOOLE_0:def 2;
then Y8 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/
Z9 by XBOOLE_0:def 2;
then Y8 in V by XBOOLE_0:def 2;
end;
then Y in Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA by A20,XBOOLE_0:def 2
;
then Y in Z3 \/ (Z4 \/ Z5) \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA by XBOOLE_1:4;
then Y in Z3 \/ (Z4 \/ Z5 \/ Z6) \/ Z7 \/ Z8 \/ Z9 \/ ZA by XBOOLE_1:4;
then Y in Z3 \/ (Z4 \/ Z5 \/ Z6 \/ Z7) \/ Z8 \/ Z9 \/ ZA by XBOOLE_1:4;
then Y in Z3 \/ (Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8) \/ Z9 \/ ZA by XBOOLE_1:4;
then Y in Z3 \/ (Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9) \/ ZA by XBOOLE_1:4;
then A23: Y in Z3 \/ (Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA) by XBOOLE_1:4;
now assume
A24:  Y in Z3;
then consider Y1,Y2,Y3,Y4,Y5,Y6,Y7 such that
A25:    Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in
Y
& Y1 meets X by A3;
Y in union union union X by A3,A24;
then Y7 in union union union union X by A25,TARSKI:def 4;
then Y7 in Z4 by A4,A25;
then Y7 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_0:def 2;
then Y7 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_0:def 2;
then Y7 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_0:def 2;
then Y7 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_0:def 2;
then Y7 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/
Z8 by XBOOLE_0:def 2;
then Y7 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/
Z9 by XBOOLE_0:def 2;
then Y7 in V by XBOOLE_0:def 2;
end;
then Y in Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA by A23,XBOOLE_0:def 2;
then Y in Z4 \/ (Z5 \/ Z6) \/ Z7 \/ Z8 \/ Z9 \/ ZA by XBOOLE_1:4;
then Y in Z4 \/ (Z5 \/ Z6 \/ Z7) \/ Z8 \/ Z9 \/ ZA by XBOOLE_1:4;
then Y in Z4 \/ (Z5 \/ Z6 \/ Z7 \/ Z8) \/ Z9 \/ ZA by XBOOLE_1:4;
then Y in Z4 \/ (Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9) \/ ZA by XBOOLE_1:4;
then A26: Y in Z4 \/ (Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA) by XBOOLE_1:4;
now assume
A27:  Y in Z4;
then consider Y1,Y2,Y3,Y4,Y5,Y6 such that
A28:    Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y
& Y1 meets X by A4;
Y in union union union union X by A4,A27;
then Y6 in union union union union union X by A28,TARSKI:def 4;
then Y6 in Z5 by A5,A28;
then Y6 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_0:def 2;
then Y6 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_0:def 2;
then Y6 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_0:def 2;
then Y6 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/
Z8 by XBOOLE_0:def 2;
then Y6 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/
Z9 by XBOOLE_0:def 2;
then Y6 in V by XBOOLE_0:def 2;
end;
then Y in Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA by A26,XBOOLE_0:def 2;
then Y in Z5 \/ (Z6 \/ Z7) \/ Z8 \/ Z9 \/ ZA by XBOOLE_1:4;
then Y in Z5 \/ (Z6 \/ Z7 \/ Z8) \/ Z9 \/ ZA by XBOOLE_1:4;
then Y in Z5 \/ (Z6 \/ Z7 \/ Z8 \/ Z9) \/ ZA by XBOOLE_1:4;
then A29: Y in Z5 \/ (Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA) by XBOOLE_1:4;
now assume
A30:  Y in Z5;
then consider Y1,Y2,Y3,Y4,Y5 such that
A31:    Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y & Y1 meets X by A5;
Y in union union union union union X by A5,A30;
then Y5 in union union union union union union X by A31,TARSKI:def 4;
then Y5 in Z6 by A6,A31;
then Y5 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_0:def 2;
then Y5 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_0:def 2;
then Y5 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/
Z8 by XBOOLE_0:def 2;
then Y5 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/
Z9 by XBOOLE_0:def 2;
then Y5 in V by XBOOLE_0:def 2;
end;
then Y in Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA by A29,XBOOLE_0:def 2;
then Y in Z6 \/ (Z7 \/ Z8) \/ Z9 \/ ZA by XBOOLE_1:4;
then Y in Z6 \/ (Z7 \/ Z8 \/ Z9) \/ ZA by XBOOLE_1:4;
then A32: Y in Z6 \/ (Z7 \/ Z8 \/ Z9 \/ ZA) by XBOOLE_1:4;
now assume
A33:  Y in Z6;
then consider Y1,Y2,Y3,Y4 such that
A34:    Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y & Y1 meets X by A6;
Y in union union union union union union X by A6,A33;
then Y4 in
union union union union union union union X by A34,TARSKI:def 4;
then Y4 in Z7 by A7,A34;
then Y4 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_0:def 2;
then Y4 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/
Z8 by XBOOLE_0:def 2;
then Y4 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/
Z9 by XBOOLE_0:def 2;
then Y4 in V by XBOOLE_0:def 2;
end;
then Y in Z7 \/ Z8 \/ Z9 \/ ZA by A32,XBOOLE_0:def 2;
then Y in Z7 \/ (Z8 \/ Z9) \/ ZA by XBOOLE_1:4;
then A35: Y in Z7 \/ (Z8 \/ Z9 \/ ZA) by XBOOLE_1:4;
now assume
A36:  Y in Z7;
then consider Y1,Y2,Y3 such that
A37:    Y1 in Y2 & Y2 in Y3 & Y3 in Y & Y1 meets X by A7;
Y in union union union union union union union X by A7,A36;
then Y3 in union union union union union union union union X
by A37,TARSKI:def 4;
then Y3 in Z8 by A8,A37;
then Y3 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/
Z8 by XBOOLE_0:def 2;
then Y3 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/
Z9 by XBOOLE_0:def 2;
then Y3 in V by XBOOLE_0:def 2;
end;
then Y in Z8 \/ Z9 \/ ZA by A35,XBOOLE_0:def 2;
then A38: Y in Z8 \/ (Z9 \/ ZA) by XBOOLE_1:4;
now assume
A39:  Y in Z8;
then consider Y1,Y2 such that
A40:    Y1 in Y2 & Y2 in Y & Y1 meets X by A8;
Y in union union union union union union union union X by A8,A39;
then Y2 in union union union union union union union union union X
by A40,TARSKI:def 4;
then Y2 in Z9 by A9,A40;
then Y2 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/
Z9 by XBOOLE_0:def 2;
then Y2 in V by XBOOLE_0:def 2;
end;
then A41: Y in Z9 \/ ZA by A38,XBOOLE_0:def 2;
now assume
A42:  Y in Z9;
then consider Y1 such that
A43:    Y1 in Y & Y1 meets X by A9;
Y in union union union union union union union union union X by A9,A42;
then Y1 in union union union union union union union union union union X
by A43,TARSKI:def 4;
then Y1 in ZA by A10,A43;
then Y1 in V by XBOOLE_0:def 2;
end;
then Y in ZA by A41,XBOOLE_0:def 2;
then Y meets X by A10;
end;

theorem Th2:
X <> {} implies
ex Y st Y in X &
for Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9,YA,YB
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
proof
defpred P1[set] means
ex Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9,YA
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 \$1 & Y1 meets X;
consider Z1 such that
A1: for Y holds Y in Z1 iff
Y in union X & P1[Y] from Separation;
defpred P2[set] means
ex Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9
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 \$1 & Y1 meets X;
consider Z2 such that
A2:  for Y holds Y in Z2 iff
Y in union union X & P2[Y] from Separation;
defpred P3[set] means
ex Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8
st Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7
& Y7 in Y8 & Y8 in \$1 & Y1 meets X;
consider Z3 such that
A3:  for Y holds Y in Z3 iff
Y in union union union X & P3[Y] from Separation;
defpred P4[set] means
ex Y1,Y2,Y3,Y4,Y5,Y6,Y7
st Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7
& Y7 in \$1 & Y1 meets X;
consider Z4 such that
A4:  for Y holds Y in Z4 iff
Y in union union union union X & P4[Y] from Separation;
defpred P5[set] means
ex Y1,Y2,Y3,Y4,Y5,Y6
st Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in \$1 &
Y1 meets X;
consider Z5 such that
A5:  for Y holds Y in Z5 iff
Y in union union union union union X & P5[Y] from Separation;
defpred P6[set] means
ex Y1,Y2,Y3,Y4,Y5
st Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in \$1 & Y1 meets X;
consider Z6 such that
A6:  for Y holds Y in Z6 iff
Y in union union union union union union X & P6[Y] from Separation;
defpred P7[set] means
ex Y1,Y2,Y3,Y4 st Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in \$1 & Y1 meets X;
consider Z7 such that
A7:  for Y holds Y in Z7 iff
Y in union union union union union union union X & P7[Y]
from Separation;
defpred P8[set] means
ex Y1,Y2,Y3 st Y1 in Y2 & Y2 in Y3 & Y3 in \$1 & Y1 meets X;
consider Z8 such that
A8:  for Y holds Y in Z8 iff
Y in union union union union union union union union X & P8[Y]
from Separation;
defpred P9[set] means
ex Y1,Y2 st Y1 in Y2 & Y2 in \$1 & Y1 meets X;
consider Z9 such that
A9:  for Y holds Y in Z9 iff
Y in union union union union union union union union union X & P9[Y]
from Separation;
defpred P10[set] means
ex Y1 st Y1 in \$1 & Y1 meets X;
consider ZA such that
A10:  for Y holds Y in ZA iff
Y in union union union union union union union union union union X &
P10[Y] from Separation;
defpred P11[set] means \$1 meets X;
consider ZB such that
A11:  for Y holds Y in ZB iff
Y in union union union union union union union union union union
(union X) & P11[Y] from Separation;

set V = (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA \/
ZB;

A12: V = X \/ (Z1 \/ Z2) \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA \/
ZB by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3) \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA \/
ZB by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4) \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA \/
ZB by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5) \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA \/
ZB by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6) \/ Z7 \/ Z8 \/ Z9 \/ ZA \/
ZB by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7) \/ Z8 \/ Z9 \/ ZA \/
ZB by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8) \/ Z9 \/ ZA \/
ZB by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9) \/ ZA \/
ZB by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA) \/
ZB by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA \/
ZB) by XBOOLE_1:4;

assume X <> {};
then V <> {} by A12,XBOOLE_1:15;
then consider Y such that
A13:  Y in V and
A14:  Y misses V by MCART_1:1;
assume
A15: not thesis;
Y in (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
or Y in ZB by A13,XBOOLE_0:def 2;
then Y in (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/
Z9 or Y in ZA
or Y in ZB by XBOOLE_0:def 2;
then Y in (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/
Z8 or Y in Z9 or Y in ZA
or Y in ZB by XBOOLE_0:def 2;
then Y in (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/
Z7 or Y in Z8 or Y in Z9
or Y in ZA or Y in ZB by XBOOLE_0:def 2;
then Y in (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/
Z6 or Y in Z7 or Y in Z8 or Y in
Z9
or Y in ZA or Y in ZB by XBOOLE_0:def 2;
then Y in (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/
Z5 or Y in Z6 or Y in Z7 or Y in Z8
or Y in Z9 or Y in ZA or Y in ZB by XBOOLE_0:def 2;
then Y in (X \/ Z1) \/ Z2 \/ Z3 \/
Z4 or Y in Z5 or Y in Z6 or Y in Z7 or Y in
Z8
or Y in Z9 or Y in ZA or Y in ZB by XBOOLE_0:def 2;
then Y in (X \/ Z1) \/ Z2 \/ Z3 or Y in Z4 or Y in Z5 or Y in Z6 or Y in
Z7
or Y in Z8 or Y in Z9 or Y in ZA or Y in ZB by XBOOLE_0:def 2;
then Y in (X \/ Z1) \/
Z2 or Y in Z3 or Y in Z4 or Y in Z5 or Y in Z6 or Y in
Z7
or Y in Z8 or Y in Z9 or Y in ZA or Y in ZB by XBOOLE_0:def 2;
then A16: Y in X \/
Z1 or Y in Z2 or Y in Z3 or Y in Z4 or Y in Z5 or Y in Z6 or Y in
Z7
or Y in Z8 or Y in Z9 or Y in ZA or Y in ZB by XBOOLE_0:def 2;
per cases by A16,XBOOLE_0:def 2;
suppose
A17:  Y in X;
then consider Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9,YA,YB such that
A18:   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 & not Y1 misses X by A15;

YB in union X & Y1 meets X by A17,A18,TARSKI:def 4;
then YB in Z1 by A1,A18;
then YB in X \/ Z1 by XBOOLE_0:def 2;
then YB in X \/ Z1 \/ Z2 by XBOOLE_0:def 2;
then YB in X \/ Z1 \/ Z2 \/ Z3 by XBOOLE_0:def 2;
then YB in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_0:def 2;
then YB in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_0:def 2;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by A18,XBOOLE_0:3;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/
Z8 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/
Z9 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA
by XBOOLE_1:70;
suppose
A19:  Y in Z1;
then consider Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9,YA such that
A20:    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 & Y1 meets X by A1;
Y in union X by A1,A19;
then YA in union union X by A20,TARSKI:def 4;
then YA in Z2 by A2,A20;
then YA in X \/ Z1 \/ Z2 by XBOOLE_0:def 2;
then Y meets X \/ Z1 \/ Z2 by A20,XBOOLE_0:3;
then Y meets X \/ Z1 \/ Z2 \/ Z3 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/
Z8 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/
Z9 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA
by XBOOLE_1:70;
suppose
A21:  Y in Z2;
then consider Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9 such that
A22:    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 Y & Y1 meets X by A2;
Y in union union X by A2,A21;
then Y9 in union union union X by A22,TARSKI:def 4;
then Y9 in Z3 by A3,A22;
then Y9 in X \/ Z1 \/ Z2 \/ Z3 by XBOOLE_0:def 2;
then Y9 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_0:def 2;
then Y9 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_0:def 2;
then Y9 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_0:def 2;
then Y9 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_0:def 2;
then Y9 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/
Z8 by XBOOLE_0:def 2;
then Y9 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/
Z9 by XBOOLE_0:def 2;
then Y9 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA by XBOOLE_0:def 2;
then Y9 in V by XBOOLE_0:def 2;
suppose
A23:  Y in Z3;
then consider Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8 such that
A24:    Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in
Y8
& Y8 in Y & Y1 meets X by A3;
Y in union union union X by A3,A23;
then Y8 in union union union union X by A24,TARSKI:def 4;
then Y8 in Z4 by A4,A24;
then Y8 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_0:def 2;
then Y8 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_0:def 2;
then Y8 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_0:def 2;
then Y8 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_0:def 2;
then Y8 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/
Z8 by XBOOLE_0:def 2;
then Y8 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/
Z9 by XBOOLE_0:def 2;
then Y8 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA by XBOOLE_0:def 2;
then Y8 in V by XBOOLE_0:def 2;
suppose
A25:  Y in Z4;
then consider Y1,Y2,Y3,Y4,Y5,Y6,Y7 such that
A26:    Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in
Y
& Y1 meets X by A4;
Y in union union union union X by A4,A25;
then Y7 in union union union union union X by A26,TARSKI:def 4;
then Y7 in Z5 by A5,A26;
then Y7 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_0:def 2;
then Y7 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_0:def 2;
then Y7 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_0:def 2;
then Y7 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/
Z8 by XBOOLE_0:def 2;
then Y7 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/
Z9 by XBOOLE_0:def 2;
then Y7 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA by XBOOLE_0:def 2;
then Y7 in V by XBOOLE_0:def 2;
suppose
A27:  Y in Z5;
then consider Y1,Y2,Y3,Y4,Y5,Y6 such that
A28:    Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y
& Y1 meets X by A5;
Y in union union union union union X by A5,A27;
then Y6 in union union union union union union X by A28,TARSKI:def 4;
then Y6 in Z6 by A6,A28;
then Y6 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_0:def 2;
then Y6 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_0:def 2;
then Y6 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/
Z8 by XBOOLE_0:def 2;
then Y6 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/
Z9 by XBOOLE_0:def 2;
then Y6 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA by XBOOLE_0:def 2;
then Y6 in V by XBOOLE_0:def 2;
suppose
A29:  Y in Z6;
then consider Y1,Y2,Y3,Y4,Y5 such that
A30:    Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y & Y1 meets X by A6;
Y in union union union union union union X by A6,A29;
then Y5 in
union union union union union union union X by A30,TARSKI:def 4;
then Y5 in Z7 by A7,A30;
then Y5 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_0:def 2;
then Y5 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/
Z8 by XBOOLE_0:def 2;
then Y5 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/
Z9 by XBOOLE_0:def 2;
then Y5 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA by XBOOLE_0:def 2;
then Y5 in V by XBOOLE_0:def 2;
suppose
A31:  Y in Z7;
then consider Y1,Y2,Y3,Y4 such that
A32:    Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y & Y1 meets X by A7;
Y in union union union union union union union X by A7,A31;
then Y4 in union union union union union union union union X
by A32,TARSKI:def 4;
then Y4 in Z8 by A8,A32;
then Y4 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/
Z8 by XBOOLE_0:def 2;
then Y4 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/
Z9 by XBOOLE_0:def 2;
then Y4 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA by XBOOLE_0:def 2;
then Y4 in V by XBOOLE_0:def 2;
suppose
A33:  Y in Z8;
then consider Y1,Y2,Y3 such that
A34:    Y1 in Y2 & Y2 in Y3 & Y3 in Y & Y1 meets X by A8;
Y in union union union union union union union union X by A8,A33;
then Y3 in union union union union union union union union union X
by A34,TARSKI:def 4;
then Y3 in Z9 by A9,A34;
then Y3 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/
Z9 by XBOOLE_0:def 2;
then Y3 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA by XBOOLE_0:def 2;
then Y3 in V by XBOOLE_0:def 2;
suppose
A35:  Y in Z9;
then consider Y1,Y2 such that
A36:    Y1 in Y2 & Y2 in Y & Y1 meets X by A9;
Y in union union union union union union union union union X by A9,A35;
then Y2 in union union union union union union union union union union X
by A36,TARSKI:def 4;
then Y2 in ZA by A10,A36;
then Y2 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA by XBOOLE_0:def 2;
then Y2 in V by XBOOLE_0:def 2;
suppose
A37:  Y in ZA;
then consider Y1 such that
A38:    Y1 in Y & Y1 meets X by A10;
Y in union union union union union union union union union union X
by A10,A37;
then Y1 in union union union union union union union union union union
(union X) by A38,TARSKI:def 4;
then Y1 in ZB by A11,A38;
then Y1 in V by XBOOLE_0:def 2;
suppose Y in ZB;
then Y meets X by A11;
end;

::
::   Tuples for n=7
::

definition
let x1,x2,x3,x4,x5,x6,x7;
func [x1,x2,x3,x4,x5,x6,x7] equals
:Def1:  [[x1,x2,x3,x4,x5,x6],x7];
correctness;
end;

theorem
Th3: [x1,x2,x3,x4,x5,x6,x7] = [[[[[[x1,x2],x3],x4],x5],x6],x7]
proof
thus [x1,x2,x3,x4,x5,x6,x7]
= [[x1,x2,x3,x4,x5,x6],x7] by Def1
.= [[[x1,x2,x3,x4,x5],x6],x7] by MCART_3:def 1
.= [[[[x1,x2,x3,x4],x5],x6],x7] by MCART_2:def 1
.= [[[[[x1,x2,x3],x4],x5],x6],x7] by MCART_1:def 4
.= [[[[[[x1,x2],x3],x4],x5],x6],x7] by MCART_1:def 3;
end;

canceled;

theorem
[x1,x2,x3,x4,x5,x6,x7] = [[x1,x2,x3,x4,x5],x6,x7]
proof
thus [x1,x2,x3,x4,x5,x6,x7]
= [[[[[[x1,x2],x3],x4],x5],x6],x7] by Th3
.= [[[[[x1,x2],x3],x4],x5],x6,x7] by MCART_1:def 3
.= [[x1,x2,x3,x4,x5],x6,x7] by MCART_2:3;
end;

theorem
[x1,x2,x3,x4,x5,x6,x7] = [[x1,x2,x3,x4],x5,x6,x7]
proof
thus [x1,x2,x3,x4,x5,x6,x7]
= [[[[[[x1,x2],x3],x4],x5],x6],x7] by Th3
.= [[[[x1,x2],x3],x4],x5,x6,x7] by MCART_1:31
.= [[x1,x2,x3,x4],x5,x6,x7] by MCART_1:31;
end;

theorem
[x1,x2,x3,x4,x5,x6,x7] = [[x1,x2,x3],x4,x5,x6,x7]
proof
thus [x1,x2,x3,x4,x5,x6,x7]
= [[[[[[x1,x2],x3],x4],x5],x6],x7] by Th3
.= [[[x1,x2],x3],x4,x5,x6,x7] by MCART_2:3
.= [[x1,x2,x3],x4,x5,x6,x7] by MCART_1:def 3;
end;

theorem
Th8: [x1,x2,x3,x4,x5,x6,x7] = [[x1,x2],x3,x4,x5,x6,x7]
proof
thus [x1,x2,x3,x4,x5,x6,x7]
= [[[[[[x1,x2],x3],x4],x5],x6],x7] by Th3
.= [[[x1,x2],x3],x4,x5,x6,x7] by MCART_2:3
.= [[x1,x2],x3,x4,x5,x6,x7] by MCART_3:7;
end;

theorem Th9:
[x1,x2,x3,x4,x5,x6,x7] = [y1,y2,y3,y4,y5,y6,y7]
implies x1 = y1 & x2 = y2 & x3 = y3 & x4 = y4 & x5 = y5 & x6 = y6
& x7 = y7
proof
assume [x1,x2,x3,x4,x5,x6,x7] = [y1,y2,y3,y4,y5,y6,y7];

then [[x1,x2,x3,x4,x5,x6],x7]
= [y1,y2,y3,y4,y5,y6,y7] by Def1
.= [[y1,y2,y3,y4,y5,y6],y7] by Def1;
then [x1,x2,x3,x4,x5,x6] = [y1,y2,y3,y4,y5,y6]
& x7 = y7 by ZFMISC_1:33;
hence thesis by MCART_3:8;
end;

theorem Th10:
X <> {} implies
ex x st x in X &
not ex x1,x2,x3,x4,x5,x6,x7
st (x1 in X or x2 in X) & x = [x1,x2,x3,x4,x5,x6,x7]
proof
assume X <> {};

then consider Y such that
A1:  Y in X and
A2:  for Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9,YA,YB
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 by Th2;

take x = Y;
thus x in X by A1;
given x1,x2,x3,x4,x5,x6,x7 such that
A3:  x1 in X or x2 in X and
A4:  x = [x1,x2,x3,x4,x5,x6,x7];

set Y1 = { x1, x2 },
Y2 = { Y1, {x1} },
Y3 = { Y2, x3 },
Y4 = { Y3, {Y2} },
Y5 = { Y4, x4 },
Y6 = { Y5, {Y4} },
Y7 = { Y6, x5 },
Y8 = { Y7, {Y6} },
Y9 = { Y8, x6 },
YA = { Y9, {Y8} },
YB = { YA, x7 };

x1 in Y1 & x2 in Y1 by TARSKI:def 2;
then A5: not Y1 misses X by A3,XBOOLE_0:3;

Y = [[[[[[x1,x2],x3],x4],x5],x6],x7] by A4,Th3
.= [[[[[ Y2,x3],x4],x5],x6],x7 ] by TARSKI:def 5
.= [[[[ Y4,x4],x5],x6],x7 ] by TARSKI:def 5
.= [[[ Y6,x5 ],x6],x7 ] by TARSKI:def 5
.= [[ Y8,x6],x7 ] by TARSKI:def 5
.= [ YA,x7 ] by TARSKI:def 5
.= { YB, { YA } } by TARSKI:def 5;

then 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 by TARSKI:def 2;
end;

::
::   Cartesian products of seven sets
::

definition
let X1,X2,X3,X4,X5,X6,X7;
func [:X1,X2,X3,X4,X5,X6,X7:] -> set equals
:Def2:  [:[: X1,X2,X3,X4,X5,X6 :],X7 :];
correctness;
end;

theorem Th11:
[:X1,X2,X3,X4,X5,X6,X7:] = [:[:[:[:[:[:X1,X2:],X3:],X4:],X5:],X6:],X7:]
proof
thus [:X1,X2,X3,X4,X5,X6,X7:]
= [:[:X1,X2,X3,X4,X5,X6:],X7:] by Def2
.= [:[:[:X1,X2,X3,X4,X5:],X6:],X7:] by MCART_3:def 2
.= [:[:[:[:X1,X2,X3,X4:],X5:],X6:],X7:] by MCART_2:def 2
.= [:[:[:[:[:X1,X2,X3:],X4:],X5:],X6:],X7:] by ZFMISC_1:def 4
.= [:[:[:[:[:[:X1,X2:],X3:],X4:],X5:],X6:],X7:] by ZFMISC_1:def 3;
end;

canceled;

theorem
[:X1,X2,X3,X4,X5,X6,X7:] = [:[:X1,X2,X3,X4,X5:],X6,X7:]
proof
thus [:X1,X2,X3,X4,X5,X6,X7:]
= [:[:[:[:[:[:X1,X2:],X3:],X4:],X5:],X6:],X7:] by Th11
.= [:[:[:[:[:X1,X2:],X3:],X4:],X5:],X6,X7:] by ZFMISC_1:def 3
.= [:[:X1,X2,X3,X4,X5:],X6,X7:] by MCART_2:9;
end;

theorem
[:X1,X2,X3,X4,X5,X6,X7:] = [:[:X1,X2,X3,X4:],X5,X6,X7:]
proof
thus [:X1,X2,X3,X4,X5,X6,X7:]
= [:[:[:[:[:[:X1,X2:],X3:],X4:],X5:],X6:],X7:] by Th11
.= [:[:[:[:X1,X2:],X3:],X4:],X5,X6,X7:] by MCART_1:53
.= [:[:X1,X2,X3,X4:],X5,X6,X7:] by MCART_1:53;
end;

theorem
[:X1,X2,X3,X4,X5,X6,X7:] = [:[:X1,X2,X3:],X4,X5,X6,X7:]
proof
thus [:X1,X2,X3,X4,X5,X6,X7:]
= [:[:[:[:[:[:X1,X2:],X3:],X4:],X5:],X6:],X7:] by Th11
.= [:[:[:X1,X2:],X3:],X4,X5,X6,X7:] by MCART_2:9
.= [:[:X1,X2,X3:],X4,X5,X6,X7:] by ZFMISC_1:def 3;
end;

theorem
Th16: [:X1,X2,X3,X4,X5,X6,X7:] = [:[:X1,X2:],X3,X4,X5,X6,X7:]
proof
thus [:X1,X2,X3,X4,X5,X6,X7:]
= [:[:[:[:[:[:X1,X2:],X3:],X4:],X5:],X6:],X7:] by Th11
.= [:[:[:X1,X2:],X3:],X4,X5,X6,X7:] by MCART_2:9
.= [:[:X1,X2:],X3,X4,X5,X6,X7:] by MCART_3:14;
end;

theorem
Th17: X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {}
& X7 <> {}
iff [:X1,X2,X3,X4,X5,X6,X7:] <> {}
proof
A1: [:[:X1,X2,X3,X4,X5,X6:],X7:] = [:X1,X2,X3,X4,X5,X6,X7:] by Def2;
X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {}
iff [:X1,X2,X3,X4,X5,X6:] <> {} by MCART_3:15;
hence thesis by A1,ZFMISC_1:113;
end;

theorem Th18:
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<> {} implies
( [:X1,X2,X3,X4,X5,X6,X7:] = [:Y1,Y2,Y3,Y4,Y5,Y6,Y7:]
implies X1=Y1 & X2=Y2 & X3=Y3 & X4=Y4 & X5=Y5 & X6=Y6 & X7=Y7 )
proof
A1: [:X1,X2,X3,X4,X5,X6,X7:] = [:[:X1,X2,X3,X4,X5,X6:],X7:] by Def2;
assume
A2: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{};
then A3: [:X1,X2,X3,X4,X5,X6:] <> {} by MCART_3:15;
assume
A4: X7<>{};
assume [:X1,X2,X3,X4,X5,X6,X7:] = [:Y1,Y2,Y3,Y4,Y5,Y6,Y7:];
then [:[:X1,X2,X3,X4,X5,X6:],X7:]
= [:[:Y1,Y2,Y3,Y4,Y5,Y6:],Y7:] by A1,Def2;
then [:X1,X2,X3,X4,X5,X6:] = [:Y1,Y2,Y3,Y4,Y5,Y6:]
& X7 = Y7 by A3,A4,ZFMISC_1:134;
hence thesis by A2,MCART_3:16;
end;

theorem
[:X1,X2,X3,X4,X5,X6,X7:]<>{}
& [:X1,X2,X3,X4,X5,X6,X7:] = [:Y1,Y2,Y3,Y4,Y5,Y6,Y7:]
implies X1=Y1 & X2=Y2 & X3=Y3 & X4=Y4 & X5=Y5 & X6=Y6 & X7=Y7
proof
assume [:X1,X2,X3,X4,X5,X6,X7:]<>{};
then X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} by Th17;
hence thesis by Th18;
end;

theorem
[:X,X,X,X,X,X,X:] = [:Y,Y,Y,Y,Y,Y,Y:] implies X = Y
proof
assume [:X,X,X,X,X,X,X:] = [:Y,Y,Y,Y,Y,Y,Y:];
then X<>{} or Y<>{} implies thesis by Th18;
hence X = Y;
end;

reserve xx7 for Element of X7;

theorem Th21:
X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {}
implies
for x being Element of [:X1,X2,X3,X4,X5,X6,X7:]
ex xx1,xx2,xx3,xx4,xx5,xx6,xx7 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7]
proof
assume
A1:  X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <> {}
;

then A2:  [:X1,X2,X3,X4,X5,X6:] <> {} by MCART_3:15;

let x be Element of [:X1,X2,X3,X4,X5,X6,X7:];
reconsider x'=x as
Element of [:[:X1,X2,X3,X4,X5,X6:],X7:] by Def2;
consider x123456 being (Element of [:X1,X2,X3,X4,X5,X6:]),
xx7 such that
A3:   x' = [x123456,xx7] by A1,A2,MCART_2:36;
consider xx1,xx2,xx3,xx4,xx5,xx6 such that
A4:   x123456 = [xx1,xx2,xx3,xx4,xx5,xx6] by A1,MCART_3:19;

take xx1,xx2,xx3,xx4,xx5,xx6,xx7;
thus x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] by A3,A4,Def1;
end;

definition
let X1,X2,X3,X4,X5,X6,X7;
assume A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{};
let x be Element of [:X1,X2,X3,X4,X5,X6,X7:];

func x`1 -> Element of X1 means
:Def3: x = [x1,x2,x3,x4,x5,x6,x7] implies it = x1;
existence
proof
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7 such that
A2:   x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] by A1,Th21;
take xx1;
thus thesis by A2,Th9;
end;
uniqueness
proof
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7 such that
A3:   x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] by A1,Th21;
let y,z be Element of X1;
assume x = [x1,x2,x3,x4,x5,x6,x7] implies y = x1;
then A4:   y = xx1 by A3;
assume x = [x1,x2,x3,x4,x5,x6,x7] implies z = x1;
hence thesis by A3,A4;
end;

func x`2 -> Element of X2 means
:Def4: x = [x1,x2,x3,x4,x5,x6,x7] implies it = x2;
existence
proof
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7 such that
A5:   x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] by A1,Th21;
take xx2; thus thesis by A5,Th9;
end;
uniqueness
proof
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7 such that
A6:   x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] by A1,Th21;
let y,z be Element of X2;
assume x = [x1,x2,x3,x4,x5,x6,x7] implies y = x2;
then A7:   y = xx2 by A6;
assume x = [x1,x2,x3,x4,x5,x6,x7] implies z = x2;
hence thesis by A6,A7;
end;

func x`3 -> Element of X3 means
:Def5: x = [x1,x2,x3,x4,x5,x6,x7] implies it = x3;
existence
proof
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7 such that
A8:   x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] by A1,Th21;
take xx3; thus thesis by A8,Th9;
end;
uniqueness
proof
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7 such that
A9:   x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] by A1,Th21;
let y,z be Element of X3;
assume x = [x1,x2,x3,x4,x5,x6,x7] implies y = x3;
then A10:   y = xx3 by A9;
assume x = [x1,x2,x3,x4,x5,x6,x7] implies z = x3;
hence thesis by A9,A10;
end;

func x`4 -> Element of X4 means
:Def6: x = [x1,x2,x3,x4,x5,x6,x7] implies it = x4;
existence
proof
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7 such that
A11:   x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] by A1,Th21;
take xx4; thus thesis by A11,Th9;
end;
uniqueness
proof
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7 such that
A12:   x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] by A1,Th21;
let y,z be Element of X4;
assume x = [x1,x2,x3,x4,x5,x6,x7] implies y = x4;
then A13:   y = xx4 by A12;
assume x = [x1,x2,x3,x4,x5,x6,x7] implies z = x4;
hence thesis by A12,A13;
end;

func x`5 -> Element of X5 means
:Def7: x = [x1,x2,x3,x4,x5,x6,x7] implies it = x5;
existence
proof
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7 such that
A14:   x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] by A1,Th21;
take xx5; thus thesis by A14,Th9;
end;
uniqueness
proof
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7 such that
A15:   x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] by A1,Th21;
let y,z be Element of X5;
assume x = [x1,x2,x3,x4,x5,x6,x7] implies y = x5;
then A16:   y = xx5 by A15;
assume x = [x1,x2,x3,x4,x5,x6,x7] implies z = x5;
hence thesis by A15,A16;
end;

func x`6 -> Element of X6 means
:Def8: x = [x1,x2,x3,x4,x5,x6,x7] implies it = x6;
existence
proof
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7 such that
A17:   x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] by A1,Th21;
take xx6; thus thesis by A17,Th9;
end;
uniqueness
proof
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7 such that
A18:   x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] by A1,Th21;
let y,z be Element of X6;
assume x = [x1,x2,x3,x4,x5,x6,x7] implies y = x6;
then A19:   y = xx6 by A18;
assume x = [x1,x2,x3,x4,x5,x6,x7] implies z = x6;
hence thesis by A18,A19;
end;

func x`7 -> Element of X7 means
:Def9: x = [x1,x2,x3,x4,x5,x6,x7] implies it = x7;
existence
proof
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7 such that
A20:   x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] by A1,Th21;
take xx7; thus thesis by A20,Th9;
end;
uniqueness
proof
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7 such that
A21:   x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] by A1,Th21;
let y,z be Element of X7;
assume x = [x1,x2,x3,x4,x5,x6,x7] implies y = x7;
then A22:   y = xx7 by A21;
assume x = [x1,x2,x3,x4,x5,x6,x7] implies z = x7;
hence thesis by A21,A22;
end;
end;

theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} implies
for x being Element of [:X1,X2,X3,X4,X5,X6,X7:]
for x1,x2,x3,x4,x5,x6,x7 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 Def3,Def4,Def5,Def6,Def7,Def8,Def9;

theorem Th23:
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} implies
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]
proof
assume
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{};

let x be Element of [:X1,X2,X3,X4,X5,X6,X7:];
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7 such that

A2:  x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] by A1,Th21;

thus x = [x`1,xx2,xx3,xx4,xx5,xx6,xx7] by A1,A2,Def3
.= [x`1,x`2,xx3,xx4,xx5,xx6,xx7] by A1,A2,Def4
.= [x`1,x`2,x`3,xx4,xx5,xx6,xx7] by A1,A2,Def5
.= [x`1,x`2,x`3,x`4,xx5,xx6,xx7] by A1,A2,Def6
.= [x`1,x`2,x`3,x`4,x`5,xx6,xx7] by A1,A2,Def7
.= [x`1,x`2,x`3,x`4,x`5,x`6,xx7] by A1,A2,Def8
.= [x`1,x`2,x`3,x`4,x`5,x`6,x`7] by A1,A2,Def9;
end;

theorem Th24:
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} implies
for x being Element of [:X1,X2,X3,X4,X5,X6,X7:] holds
x`1 = (x qua set)`1`1`1`1`1`1 &
x`2 = (x qua set)`1`1`1`1`1`2 &
x`3 = (x qua set)`1`1`1`1`2 &
x`4 = (x qua set)`1`1`1`2 &
x`5 = (x qua set)`1`1`2 &
x`6 = (x qua set)`1`2 &
x`7 = (x qua set)`2
proof
assume
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{};

let x be Element of [:X1,X2,X3,X4,X5,X6,X7:];

thus x`1
= [ x`1, x`2]`1 by MCART_1:7
.= [[x`1, x`2],x`3]`1`1 by MCART_1:7
.= [ x`1, x`2 ,x`3]`1`1 by MCART_1:def 3
.= [[x`1, x`2 ,x`3],x`4]`1`1`1 by MCART_1:7
.= [ x`1, x`2 ,x`3 ,x`4]`1`1`1 by MCART_1:def 4
.= [[x`1, x`2 ,x`3 ,x`4], x`5]`1`1`1`1 by MCART_1:7
.= [ x`1, x`2 ,x`3 ,x`4 , x`5]`1`1`1`1 by MCART_2:def 1
.= [[x`1, x`2 ,x`3 ,x`4, x`5],x`6]`1`1`1`1`1 by MCART_1:7
.= [ x`1, x`2 ,x`3 ,x`4 ,x`5, x`6]`1`1`1`1`1 by MCART_3:def 1
.= [[x`1, x`2 ,x`3 ,x`4, x`5, x`6],x`7]`1`1`1`1`1`1 by MCART_1:7
.= [ x`1, x`2 ,x`3 ,x`4 ,x`5, x`6, x`7]`1`1`1`1`1`1 by Def1
.= (x qua set)`1`1`1`1`1`1 by A1,Th23;

thus x`2
= [ x`1, x`2]`2 by MCART_1:7
.= [[x`1, x`2],x`3]`1`2 by MCART_1:7
.= [ x`1, x`2, x`3]`1`2 by MCART_1:def 3
.= [[x`1, x`2, x`3],x`4]`1`1`2 by MCART_1:7
.= [ x`1, x`2, x`3, x`4]`1`1`2 by MCART_1:def 4
.= [[x`1, x`2, x`3, x`4 ], x`5]`1`1`1`2 by MCART_1:7
.= [ x`1, x`2, x`3, x`4 , x`5]`1`1`1`2 by MCART_2:def 1
.= [[x`1, x`2, x`3, x`4, x`5],x`6]`1`1`1`1`2 by MCART_1:7
.= [ x`1, x`2, x`3, x`4, x`5, x`6]`1`1`1`1`2 by MCART_3:def 1
.= [[x`1, x`2, x`3, x`4, x`5, x`6],x`7]`1`1`1`1`1`2 by MCART_1:7
.= [ x`1, x`2, x`3, x`4, x`5, x`6, x`7]`1`1`1`1`1`2 by Def1
.= (x qua set)`1`1`1`1`1`2 by A1,Th23;

thus x`3
= [[x`1, x`2],x`3]`2 by MCART_1:7
.= [ x`1, x`2, x`3]`2 by MCART_1:def 3
.= [[x`1, x`2, x`3],x`4]`1`2 by MCART_1:7
.= [ x`1, x`2, x`3, x`4]`1`2 by MCART_1:def 4
.= [[x`1, x`2, x`3, x`4],x`5]`1`1`2 by MCART_1:7
.= [ x`1, x`2, x`3, x`4 ,x`5]`1`1`2 by MCART_2:def 1
.= [[x`1, x`2, x`3, x`4, x`5],x`6]`1`1`1`2 by MCART_1:7
.= [ x`1, x`2, x`3, x`4, x`5, x`6]`1`1`1`2 by MCART_3:def 1
.= [[x`1, x`2, x`3, x`4, x`5, x`6],x`7]`1`1`1`1`2 by MCART_1:7
.= [ x`1, x`2, x`3, x`4, x`5, x`6, x`7]`1`1`1`1`2 by Def1
.= (x qua set)`1`1`1`1`2 by A1,Th23;

thus x`4
= [[x`1,x`2,x`3],x`4]`2 by MCART_1:7
.= [ x`1,x`2,x`3, x`4]`2 by MCART_1:def 4
.= [[x`1,x`2,x`3, x`4],x`5]`1`2 by MCART_1:7
.= [ x`1,x`2,x`3, x`4 ,x`5]`1`2 by MCART_2:def 1
.= [[x`1,x`2,x`3, x`4, x`5],x`6]`1`1`2 by MCART_1:7
.= [ x`1,x`2,x`3, x`4, x`5, x`6]`1`1`2 by MCART_3:def 1
.= [[x`1,x`2,x`3, x`4, x`5, x`6],x`7]`1`1`1`2 by MCART_1:7
.= [ x`1,x`2,x`3, x`4, x`5, x`6, x`7]`1`1`1`2 by Def1
.= (x qua set)`1`1`1`2 by A1,Th23;

thus x`5
= [[x`1,x`2,x`3,x`4],x`5]`2 by MCART_1:7
.= [ x`1,x`2,x`3,x`4 ,x`5]`2 by MCART_2:def 1
.= [[x`1,x`2,x`3,x`4,x`5],x`6]`1`2 by MCART_1:7
.= [ x`1,x`2,x`3,x`4,x`5, x`6]`1`2 by MCART_3:def 1
.= [[x`1,x`2,x`3,x`4,x`5,x`6],x`7]`1`1`2 by MCART_1:7
.= [ x`1,x`2,x`3,x`4,x`5, x`6,x`7]`1`1`2 by Def1
.= (x qua set)`1`1`2 by A1,Th23;

thus x`6
= [[x`1,x`2,x`3,x`4,x`5],x`6]`2 by MCART_1:7
.= [ x`1,x`2,x`3,x`4,x`5, x`6]`2 by MCART_3:def 1
.= [[x`1,x`2,x`3,x`4,x`5,x`6],x`7]`1`2 by MCART_1:7
.= [ x`1,x`2,x`3,x`4,x`5,x`6, x`7]`1`2 by Def1
.= (x qua set)`1`2 by A1,Th23;

thus x`7
= [[x`1,x`2,x`3,x`4,x`5,x`6],x`7]`2 by MCART_1:7
.= [ x`1,x`2,x`3,x`4,x`5,x`6, x`7]`2 by Def1
.= (x qua set)`2 by A1,Th23;
end;

theorem
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:]
implies X1 = {}
proof
assume that
A1:  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:]
and
A2: X1 <> {};
[:X1,X2,X3,X4,X5,X6,X7:]<>{}
or [:X2,X3,X4,X5,X6,X7,X1:]<>{}
or [:X3,X4,X5,X6,X7,X1,X2:]<>{}
or [:X4,X5,X6,X7,X1,X2,X3:]<>{}
or [:X5,X6,X7,X1,X2,X3,X4:]<>{}
or [:X6,X7,X1,X2,X3,X4,X5:]<>{}
or [:X7,X1,X2,X3,X4,X5,X6:]<>{} by A1,A2,XBOOLE_1:3;

then A3: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} by Th17;
now per cases by A1;
suppose
A4:  X1 c= [:X1,X2,X3,X4,X5,X6,X7:];
consider y such that
A5:   y in X1 and
A6:   for x1,x2,x3,x4,x5,x6,x7
st x1 in X1 or x2 in X1 holds y <> [x1,x2,x3,x4,x5,x6,x7]
by A2,Th10;
reconsider y as Element of [:X1,X2,X3,X4,X5,X6,X7:] by A4,A5;
y = [y`1,y`2,y`3,y`4,y`5,y`6,y`7]
& y`1 in X1 by A3,Th23;

suppose X1 c= [:X2,X3,X4,X5,X6,X7,X1:];
then X1 c= [:[:X2,X3:],X4,X5,X6,X7,X1:] by Th16;
hence thesis by A2,MCART_3:23;

suppose X1 c= [:X3,X4,X5,X6,X7,X1,X2:];
then X1 c= [:[:X3,X4:],X5,X6,X7,X1,X2:] by Th16;
hence thesis by A2,MCART_3:23;

suppose X1 c= [:X4,X5,X6,X7,X1,X2,X3:];
then X1 c= [:[:X4,X5:],X6,X7,X1,X2,X3:] by Th16;
hence thesis by A2,MCART_3:23;

suppose X1 c= [:X5,X6,X7,X1,X2,X3,X4:];
then X1 c= [:[:X5,X6:],X7,X1,X2,X3,X4:] by Th16;
hence thesis by A2,MCART_3:23;

suppose X1 c= [:X6,X7,X1,X2,X3,X4,X5:];
then X1 c= [:[:X6,X7:],X1,X2,X3,X4,X5:] by Th16;
hence thesis by A2,MCART_3:23;

suppose
A7:  X1 c= [:X7,X1,X2,X3,X4,X5,X6:];
consider y such that
A8:   y in X1 and
A9:   for x1,x2,x3,x4,x5,x6,x7
st x1 in X1 or x2 in X1 holds y <> [x1,x2,x3,x4,x5,x6,x7] by A2,Th10;
reconsider y as Element of [:X7,X1,X2,X3,X4,X5,X6:] by A7,A8;
y = [y`1,y`2,y`3,y`4,y`5,y`6,y`7]
& y`2 in X1 by A3,Th23;
hence thesis by A9;
end;
end;

theorem
[:X1,X2,X3,X4,X5,X6,X7:] meets [:Y1,Y2,Y3,Y4,Y5,Y6,Y7:] implies
X1 meets Y1 & X2 meets Y2 & X3 meets Y3 & X4 meets Y4 & X5 meets Y5
& X6 meets Y6 & X7 meets Y7
proof
A1: [:[:[:[:[:[:X1,X2:],X3:],X4:],X5:],X6:],X7:]
= [:X1,X2,X3,X4,X5,X6,X7:]
& [:[:[:[:[:[:Y1,Y2:],Y3:],Y4:],Y5:],Y6:],Y7:]
= [:Y1,Y2,Y3,Y4,Y5,Y6,Y7:] by Th11;
assume [:X1,X2,X3,X4,X5,X6,X7:] meets [:Y1,Y2,Y3,Y4,Y5,Y6,Y7:];
then A2: [:[:[:[:[:X1,X2:],X3:],X4:],X5:],X6:] meets
[:[:[:[:[:Y1,Y2:],Y3:],Y4:],Y5:],Y6:] & X7 meets Y7 by A1,ZFMISC_1:127;
then [:[:[:[:X1,X2:],X3:],X4:],X5:] meets
[:[:[:[:Y1,Y2:],Y3:],Y4:],Y5:] & X6 meets Y6 by ZFMISC_1:127;
then A3: [:[:[:X1,X2:],X3:],X4:] meets [:[:[:Y1,Y2:],Y3:],Y4:]
& X5 meets Y5 by ZFMISC_1:127;
then [:[:X1,X2:],X3:] meets [:[:Y1,Y2:],Y3:] & X4 meets Y4
by ZFMISC_1:127;
then [:X1,X2:] meets [:Y1,Y2:] & X3 meets Y3 by ZFMISC_1:127;
hence thesis by A2,A3,ZFMISC_1:127;
end;

theorem [:{x1},{x2},{x3},{x4},{x5},{x6},{x7}:]
= { [x1,x2,x3,x4,x5,x6,x7] }
proof thus
[:{x1},{x2},{x3},{x4},{x5},{x6},{x7}:]
= [:[:{x1},{x2}:],{x3},{x4},{x5},{x6},{x7}:] by Th16
.= [:{[x1,x2]}, {x3},{x4},{x5},{x6},{x7}:] by ZFMISC_1:35
.= { [[x1,x2], x3, x4, x5, x6, x7]} by MCART_3:25
.= { [x1,x2,x3,x4,x5,x6,x7] } by Th8;
end;

reserve A1 for Subset of X1,
A2 for Subset of X2,
A3 for Subset of X3,
A4 for Subset of X4,
A5 for Subset of X5,
A6 for Subset of X6,
A7 for Subset of X7;

:: 7 - Tuples

reserve x for Element of [:X1,X2,X3,X4,X5,X6,X7:];

theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} implies
for x1,x2,x3,x4,x5,x6,x7 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 Def3,Def4,Def5,Def6,Def7,Def8,Def9;

theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} &
(for xx1,xx2,xx3,xx4,xx5,xx6,xx7
st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] holds y1 = xx1)
implies y1 =x`1
proof
assume that
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} and
A2:     for xx1,xx2,xx3,xx4,xx5,xx6,xx7
st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] holds y1 = xx1;
x = [x`1,x`2,x`3,x`4,x`5,x`6,x`7] by A1,Th23;
hence thesis by A2;
end;

theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} &
(for xx1,xx2,xx3,xx4,xx5,xx6,xx7
st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] holds y2 = xx2)
implies y2 =x`2
proof
assume that
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} and
A2:    for xx1,xx2,xx3,xx4,xx5,xx6,xx7
st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] holds y2 = xx2;
x = [x`1,x`2,x`3,x`4,x`5,x`6,x`7] by A1,Th23;
hence thesis by A2;
end;

theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} &
(for xx1,xx2,xx3,xx4,xx5,xx6,xx7
st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] holds y3 = xx3)
implies y3 =x`3
proof
assume that
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} and
A2:  for xx1,xx2,xx3,xx4,xx5,xx6,xx7 st
x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] holds y3 = xx3;
x = [x`1,x`2,x`3,x`4,x`5,x`6,x`7] by A1,Th23;
hence thesis by A2;
end;

theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} &
(for xx1,xx2,xx3,xx4,xx5,xx6,xx7
st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] holds y4 = xx4)
implies y4 =x`4
proof
assume that
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} and
A2:    for xx1,xx2,xx3,xx4,xx5,xx6,xx7
st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] holds y4 = xx4;
x = [x`1,x`2,x`3,x`4,x`5,x`6,x`7] by A1,Th23;
hence thesis by A2;
end;

theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} &
(for xx1,xx2,xx3,xx4,xx5,xx6,xx7
st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] holds y5 = xx5)
implies y5 =x`5
proof
assume that
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} and
A2:    for xx1,xx2,xx3,xx4,xx5,xx6,xx7
st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] holds y5 = xx5;
x = [x`1,x`2,x`3,x`4,x`5,x`6,x`7] by A1,Th23;
hence thesis by A2;
end;

theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} &
(for xx1,xx2,xx3,xx4,xx5,xx6,xx7
st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] holds y6 = xx6)
implies y6 =x`6
proof
assume that
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} and
A2:    for xx1,xx2,xx3,xx4,xx5,xx6,xx7
st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] holds y6 = xx6;
x = [x`1,x`2,x`3,x`4,x`5,x`6,x`7] by A1,Th23;
hence thesis by A2;
end;

theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} &
(for xx1,xx2,xx3,xx4,xx5,xx6,xx7
st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] holds y7 = xx7)
implies y7 =x`7
proof
assume that
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} and
A2:    for xx1,xx2,xx3,xx4,xx5,xx6,xx7
st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] holds y7 = xx7;
x = [x`1,x`2,x`3,x`4,x`5,x`6,x`7] by A1,Th23;
hence thesis by A2;
end;

theorem
y in [: X1,X2,X3,X4,X5,X6,X7 :] implies
ex x1,x2,x3,x4,x5,x6,x7
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]
proof
assume y in [: X1,X2,X3,X4,X5,X6,X7 :];
then y in [:[:X1,X2,X3,X4,X5,X6:],X7:] by Def2;
then consider x123456, x7 being set such that
A1:   x123456 in [:X1,X2,X3,X4,X5,X6:] and
A2:   x7 in X7 and
A3:   y = [x123456,x7] by ZFMISC_1:def 2;
consider x1, x2, x3, x4, x5, x6 such that
A4:   x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5 & x6 in X6
& x123456 = [x1,x2,x3,x4,x5,x6] by A1,MCART_3:33;
y = [x1,x2,x3,x4,x5,x6,x7] by A3,A4,Def1;
hence thesis by A2,A4;
end;

theorem
[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
proof
A1: [:X1,X2,X3,X4,X5,X6,X7:] = [:[:X1,X2:],X3,X4,X5,X6,X7:] by Th16;
A2: [x1,x2,x3,x4,x5,x6,x7] = [[x1,x2],x3,x4,x5,x6,x7] by Th8;
[x1,x2] in [:X1,X2:] iff x1 in X1 & x2 in X2 by ZFMISC_1:106;
hence thesis by A1,A2,MCART_3:34;
end;

theorem
(for y holds y in Z iff
ex x1,x2,x3,x4,x5,x6,x7
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])
implies Z = [: X1,X2,X3,X4,X5,X6,X7 :]
proof
assume
A1:   for y holds y in Z iff
ex x1,x2,x3,x4,x5,x6,x7
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];
now let y;
thus y in Z implies y in [:[:X1,X2,X3,X4,X5,X6:],X7:]
proof
assume y in Z; then consider x1,x2,x3,x4,x5,x6,x7 such that
A2:      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] by A1;
A3:      y = [[x1,x2,x3,x4,x5,x6],x7] by A2,Def1;
[x1,x2,x3,x4,x5,x6] in [:X1,X2,X3,X4,X5,X6:]
& x7 in X7 by A2,MCART_3:34;
hence y in [:[:X1,X2,X3,X4,X5,X6:],X7:] by A3,ZFMISC_1:def 2;
end;
assume y in [:[:X1,X2,X3,X4,X5,X6:],X7:];
then consider x123456,x7 being set such that
A4:     x123456 in [:X1,X2,X3,X4,X5,X6:] and
A5:     x7 in X7 and
A6:     y = [x123456,x7] by ZFMISC_1:def 2;
consider x1,x2,x3,x4,x5,x6 such that
A7:   x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5 & x6 in X6
& x123456 = [x1,x2,x3,x4,x5,x6] by A4,MCART_3:33;
y = [x1,x2,x3,x4,x5,x6,x7] by A6,A7,Def1;
hence y in Z by A1,A5,A7;
end;
then Z = [:[:X1,X2,X3,X4,X5,X6:],X7:] by TARSKI:2;
hence Z = [: X1,X2,X3,X4,X5,X6,X7 :] by Def2;
end;

theorem Th39:
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{}
& Y1<>{} & Y2<>{} & Y3<>{} & Y4<>{} & Y5<>{} & Y6<>{} & Y7<>{} implies
for x being (Element of [:X1,X2,X3,X4,X5,X6,X7:]),
y being Element of [:Y1,Y2,Y3,Y4,Y5,Y6,Y7:]
holds x = y
implies 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
proof
assume that
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} and
A2: Y1<>{} & Y2<>{} & Y3<>{} & Y4<>{} & Y5<>{} & Y6<>{} & Y7<>{};
let x be Element of [:X1,X2,X3,X4,X5,X6,X7:];
let y be Element of [:Y1,Y2,Y3,Y4,Y5,Y6,Y7:];
assume
A3: x = y;
thus x`1 = (x qua set)`1`1`1`1`1`1 by A1,Th24 .= y`1 by A2,A3,Th24;
thus x`2 = (x qua set)`1`1`1`1`1`2 by A1,Th24 .= y`2 by A2,A3,Th24;
thus x`3 = (x qua set)`1`1`1`1`2 by A1,Th24 .= y`3 by A2,A3,Th24;
thus x`4 = (x qua set)`1`1`1`2 by A1,Th24 .= y`4 by A2,A3,Th24;
thus x`5 = (x qua set)`1`1`2 by A1,Th24 .= y`5 by A2,A3,Th24;
thus x`6 = (x qua set)`1`2 by A1,Th24 .= y`6 by A2,A3,Th24;
thus x`7 = (x qua set)`2 by A1,Th24 .= y`7 by A2,A3,Th24;
end;

theorem
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
proof
let x be Element of [:X1,X2,X3,X4,X5,X6,X7:];
assume
A1: x in [:A1,A2,A3,A4,A5,A6,A7:];
then A2: A1<>{} & A2<>{} & A3<>{} & A4<>{} & A5<>{} & A6<>{} & A7<>{} by
Th17;
reconsider y = x as Element of [:A1,A2,A3,A4,A5,A6,A7:] by A1;
y`1 in A1 & y`2 in A2 & y`3 in A3 & y`4 in A4 & y`5 in A5
& y`6 in A6 & y`7 in A7 by A2;
hence 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 by Th39;
end;

theorem Th41:
X1 c= Y1 & X2 c= Y2 & X3 c= Y3 & X4 c= Y4 & X5 c= Y5 & X6 c= Y6 & X7 c= Y7
implies [:X1,X2,X3,X4,X5,X6,X7:] c= [:Y1,Y2,Y3,Y4,Y5,Y6,Y7:]
proof
assume X1 c= Y1 & X2 c= Y2 & X3 c= Y3 & X4 c= Y4 & X5 c= Y5 & X6 c= Y6;
then A1: [:X1,X2,X3,X4,X5,X6:] c= [:Y1,Y2,Y3,Y4,Y5,Y6:] by MCART_3:38;
A2: [:X1,X2,X3,X4,X5,X6,X7:] = [:[:X1,X2,X3,X4,X5,X6:],X7:] &
[:Y1,Y2,Y3,Y4,Y5,Y6,Y7:] = [:[:Y1,Y2,Y3,Y4,Y5,Y6:],Y7:] by Def2;
assume X7 c= Y7;
hence thesis by A1,A2,ZFMISC_1:119;
end;

theorem
[:A1,A2,A3,A4,A5,A6,A7:] is Subset of [:X1,X2,X3,X4,X5,X6,X7:] by Th41;

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