### \$N\$-Tuples and Cartesian Products for \$n=5\$

by
Michal Muzalewski, and
Wojciech Skaba

Copyright (c) 1990 Association of Mizar Users

MML identifier: MCART_2
[ MML identifier index ]

```environ

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

begin

reserve x,x1,x2,x3,x4,x5, y,y1,y2,y3,y4,y5,
X,X1,X2,X3,X4,X5, Y,Y1,Y2,Y3,Y4,Y5,Y6,Y7,
Z,Z1,Z2,Z3,Z4,Z5,Z6,Z7 for set;
reserve xx1 for Element of X1, xx2 for Element of X2,
xx3 for Element of X3, xx4 for Element of X4,
xx5 for Element of X5;

Lm1:
X1 <> {} & X2 <> {} implies
for x being Element of [:X1,X2:]
ex xx1 being (Element of X1),
xx2 being Element of X2 st x = [xx1,xx2]
proof
assume
A1:  X1 <> {} & X2 <> {};
then A2:  [:X1,X2:] <> {} by ZFMISC_1:113;
let x be Element of [:X1,X2:];
reconsider xx1 = x`1 as Element of X1 by A2,MCART_1:10;
reconsider xx2 = x`2 as Element of X2 by A2,MCART_1:10;
take xx1,xx2;
thus x = [xx1,xx2] by A1,MCART_1:24;
end;

Lm2:
X1 <> {} & X2 <> {} & X3 <> {} implies
for x being Element of [:X1,X2,X3:]
ex xx1,xx2,xx3 st x = [xx1,xx2,xx3]
proof
assume
A1:  X1 <> {} & X2 <> {} & X3 <> {};
then A2:  [:X1,X2:] <> {} by ZFMISC_1:113;
let x be Element of [:X1,X2,X3:];
reconsider x'=x as Element of [:[:X1,X2:],X3:] by ZFMISC_1:def 3;
consider x12 being (Element of [:X1,X2:]), xx3 such that
A3:    x' = [x12,xx3] by A1,A2,Lm1;
consider xx1,xx2 such that
A4:    x12 = [xx1,xx2] by A1,Lm1;
take xx1,xx2,xx3;
thus x = [xx1,xx2,xx3] by A3,A4,MCART_1:def 3;
end;

Lm3:
X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} implies
for x being Element of [:X1,X2,X3,X4:]
ex xx1,xx2,xx3,xx4 st x = [xx1,xx2,xx3,xx4]
proof
assume
A1:  X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {};
then A2:  [:X1,X2,X3:] <> {} by MCART_1:35;
let x be Element of [:X1,X2,X3,X4:];
reconsider x'=x as Element of [:[:X1,X2,X3:],X4:] by ZFMISC_1:def 4;
consider x123 being (Element of [:X1,X2,X3:]), xx4 such that
A3:   x' = [x123,xx4] by A1,A2,Lm1;
consider xx1,xx2,xx3 such that
A4:   x123 = [xx1,xx2,xx3] by A1,Lm2;
take xx1,xx2,xx3,xx4;
thus x = [xx1,xx2,xx3,xx4] by A3,A4,MCART_1:def 4;
end;

theorem
X <> {} implies
ex Y st Y in X &
for Y1,Y2,Y3,Y4,Y5,Y6
st Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y
holds Y1 misses X
proof
defpred P[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 Z1 such that
A1: for Y holds Y in Z1 iff Y in union X & P[Y] from Separation;
defpred Q[set] means
ex Y1,Y2,Y3,Y4 st Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in \$1 & Y1 meets X;
consider Z2 such that
A2:  for Y holds Y in Z2 iff Y in union union X & Q[Y] from Separation;
defpred R[set] means
ex Y1,Y2,Y3 st Y1 in Y2 & Y2 in Y3 & Y3 in \$1 & Y1 meets X;
consider Z3 such that
A3:  for Y holds Y in Z3 iff Y in union union union X & R[Y]
from Separation;
defpred S[set] means ex Y1,Y2 st Y1 in Y2 & Y2 in \$1 & Y1 meets X;
consider Z4 such that
A4:  for Y holds Y in Z4 iff
Y in union union union union X & S[Y] from Separation;
defpred T[set] means ex Y1 st Y1 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 & T[Y] from Separation;
defpred U[set] means \$1 meets X;
consider Z6 such that
A6:  for Y holds Y in Z6 iff
Y in union union union union union union X & U[Y] from Separation;
set V = X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6;
A7: V = X \/ (Z1 \/ Z2) \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3) \/ Z4 \/ Z5 \/ Z6 by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4) \/ Z5 \/ Z6 by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5) \/ Z6 by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6) by XBOOLE_1:4;
assume X <> {};
then V <> {} by A7,XBOOLE_1:15;
then consider Y such that
A8:  Y in V and
A9:  Y misses V by MCART_1:1;
assume
A10: not thesis;
now assume
A11:  Y in X;
then consider Y1,Y2,Y3,Y4,Y5,Y6 such that
A12:   Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y
& not Y1 misses X by A10;
Y6 in union X & Y1 meets X by A11,A12,TARSKI:def 4;
then Y6 in Z1 by A1,A12;
then Y6 in X \/ Z1 by XBOOLE_0:def 2;
then Y6 in X \/ Z1 \/ Z2 by XBOOLE_0:def 2;
then Y6 in X \/ Z1 \/ Z2 \/ Z3 by XBOOLE_0:def 2;
then Y6 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_0:def 2;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by A12,XBOOLE_0:3;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_1:70;
end;
then Y in Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by A7,A8,XBOOLE_0:def 2;
then Y in Z1 \/ (Z2 \/ Z3) \/ Z4 \/ Z5 \/ Z6 by XBOOLE_1:4;
then Y in Z1 \/ (Z2 \/ Z3 \/ Z4) \/ Z5 \/ Z6 by XBOOLE_1:4;
then Y in Z1 \/ (Z2 \/ Z3 \/ Z4 \/ Z5) \/ Z6 by XBOOLE_1:4;
then A13: Y in Z1 \/ (Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6) by XBOOLE_1:4;
now assume
A14:  Y in Z1;
then consider Y1,Y2,Y3,Y4,Y5 such that
A15:    Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y & Y1 meets X by A1;
Y in union X by A1,A14;
then Y5 in union union X by A15,TARSKI:def 4;
then Y5 in Z2 by A2,A15;
then Y5 in X \/ Z1 \/ Z2 by XBOOLE_0:def 2;
then Y meets X \/ Z1 \/ Z2 by A15,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;
end;
then Y in Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by A13,XBOOLE_0:def 2;
then Y in Z2 \/ (Z3 \/ Z4) \/ Z5 \/ Z6 by XBOOLE_1:4;
then Y in Z2 \/ (Z3 \/ Z4 \/ Z5) \/ Z6 by XBOOLE_1:4;
then A16: Y in Z2 \/ (Z3 \/ Z4 \/ Z5 \/ Z6) by XBOOLE_1:4;
now assume
A17:  Y in Z2;
then consider Y1,Y2,Y3,Y4 such that
A18:    Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y & Y1 meets X by A2;
Y in union union X by A2,A17;
then Y4 in union union union X by A18,TARSKI:def 4;
then Y4 in Z3 by A3,A18;
then Y4 in X \/ Z1 \/ Z2 \/ Z3 by XBOOLE_0:def 2;
then Y4 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_0:def 2;
then Y4 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_0:def 2;
then Y4 in V by XBOOLE_0:def 2;
end;
then Y in Z3 \/ Z4 \/ Z5 \/ Z6 by A16,XBOOLE_0:def 2;
then Y in Z3 \/ (Z4 \/ Z5) \/ Z6 by XBOOLE_1:4;
then A19: Y in Z3 \/ (Z4 \/ Z5 \/ Z6) by XBOOLE_1:4;
now assume
A20:  Y in Z3;
then consider Y1,Y2,Y3 such that
A21:    Y1 in Y2 & Y2 in Y3 & Y3 in Y & Y1 meets X by A3;
Y in union union union X by A3,A20;
then Y3 in union union union union X by A21,TARSKI:def 4;
then Y3 in Z4 by A4,A21;
then Y3 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_0:def 2;
then Y3 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_0:def 2;
then Y3 in V by XBOOLE_0:def 2;
end;
then Y in Z4 \/ Z5 \/ Z6 by A19,XBOOLE_0:def 2;
then A22: Y in Z4 \/ (Z5 \/ Z6) by XBOOLE_1:4;
now assume
A23:  Y in Z4;
then consider Y1,Y2 such that
A24:    Y1 in Y2 & Y2 in Y & Y1 meets X by A4;
Y in union union union union X by A4,A23;
then Y2 in union union union union union X by A24,TARSKI:def 4;
then Y2 in Z5 by A5,A24;
then Y2 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_0:def 2;
then Y2 in V by XBOOLE_0:def 2;
end;
then A25: Y in Z5 \/ Z6 by A22,XBOOLE_0:def 2;
now assume
A26:  Y in Z5;
then consider Y1 such that
A27:    Y1 in Y & Y1 meets X by A5;
Y in union union union union union X by A5,A26;
then Y1 in union union union union union union X by A27,TARSKI:def 4;
then Y1 in Z6 by A6,A27;
then Y1 in V by XBOOLE_0:def 2;
end;
then Y in Z6 by A25,XBOOLE_0:def 2;
then Y meets X by A6;
end;

theorem
Th2:
X <> {} implies
ex Y st Y in X &
for 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
Y
holds Y1 misses X
proof
defpred P[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 Z1 such that
A1: for Y holds Y in Z1 iff Y in union X & P[Y] from Separation;
defpred Q[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 Z2 such that
A2:  for Y holds Y in Z2 iff Y in union union X & Q[Y] from Separation;
defpred R[set] means
ex Y1,Y2,Y3,Y4 st Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in \$1 & Y1 meets X;
consider Z3 such that
A3:  for Y holds Y in Z3 iff Y in union union union X & R[Y] from Separation;
defpred S[set] means
ex Y1,Y2,Y3 st Y1 in Y2 & Y2 in Y3 & Y3 in \$1 & Y1 meets X;
consider Z4 such that
A4:  for Y holds Y in Z4 iff
Y in union union union union X & S[Y] from Separation;
defpred T[set] means
ex Y1,Y2 st Y1 in Y2 & Y2 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 & T[Y] from Separation;
defpred U[set] means ex Y1 st Y1 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 & U[Y] from Separation;
defpred V[set] means \$1 meets X;
consider Z7 such that
A7:  for Y holds Y in Z7 iff
Y in union union union union union union union X & V[Y]
from Separation;
set V = X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7;
A8: V = X \/ (Z1 \/ Z2) \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3) \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4) \/ Z5 \/ Z6 \/ Z7 by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5) \/ Z6 \/ Z7 by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6) \/ Z7 by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7) by XBOOLE_1:4;
assume X <> {};
then V <> {} by A8,XBOOLE_1:15;
then consider Y such that
A9:  Y in V and
A10:  Y misses V by MCART_1:1;
assume
A11: not thesis;
now assume
A12:  Y in X;
then consider Y1,Y2,Y3,Y4,Y5,Y6,Y7 such that
A13:   Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in
Y
& not Y1 misses X by A11;

Y7 in union X & Y1 meets X by A12,A13,TARSKI:def 4;
then Y7 in Z1 by A1,A13;
then Y7 in X \/ Z1 by XBOOLE_0:def 2;
then Y7 in X \/ Z1 \/ Z2 by XBOOLE_0:def 2;
then Y7 in X \/ Z1 \/ Z2 \/ Z3 by XBOOLE_0:def 2;
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 Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by A13,XBOOLE_0:3;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_1:70;
end;
then Y in Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by A8,A9,XBOOLE_0:def 2;
then Y in Z1 \/ (Z2 \/ Z3) \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_1:4;
then Y in Z1 \/ (Z2 \/ Z3 \/ Z4) \/ Z5 \/ Z6 \/ Z7 by XBOOLE_1:4;
then Y in Z1 \/ (Z2 \/ Z3 \/ Z4 \/ Z5) \/ Z6 \/ Z7 by XBOOLE_1:4;
then Y in Z1 \/ (Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6) \/ Z7 by XBOOLE_1:4;
then A14: Y in Z1 \/ (Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7) by XBOOLE_1:4;
now assume
A15:  Y in Z1;
then consider Y1,Y2,Y3,Y4,Y5,Y6 such that
A16:    Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y
& Y1 meets X by A1;
Y in union X by A1,A15;
then Y6 in union union X by A16,TARSKI:def 4;
then Y6 in Z2 by A2,A16;
then Y6 in X \/ Z1 \/ Z2 by XBOOLE_0:def 2;
then Y meets X \/ Z1 \/ Z2 by A16,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;
end;
then Y in Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by A14,XBOOLE_0:def 2;
then Y in Z2 \/ (Z3 \/ Z4) \/ Z5 \/ Z6 \/ Z7 by XBOOLE_1:4;
then Y in Z2 \/ (Z3 \/ Z4 \/ Z5) \/ Z6 \/ Z7 by XBOOLE_1:4;
then Y in Z2 \/ (Z3 \/ Z4 \/ Z5 \/ Z6) \/ Z7 by XBOOLE_1:4;
then A17: Y in Z2 \/ (Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7) by XBOOLE_1:4;
now assume
A18:  Y in Z2;
then consider Y1,Y2,Y3,Y4,Y5 such that
A19:    Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y & Y1 meets X by A2;
Y in union union X by A2,A18;
then Y5 in union union union X by A19,TARSKI:def 4;
then Y5 in Z3 by A3,A19;
then Y5 in X \/ Z1 \/ Z2 \/ Z3 by XBOOLE_0:def 2;
then Y5 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_0:def 2;
then Y5 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_0:def 2;
then Y5 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_0:def 2;
then Y5 in V by XBOOLE_0:def 2;
end;
then Y in Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by A17,XBOOLE_0:def 2;
then Y in Z3 \/ (Z4 \/ Z5) \/ Z6 \/ Z7 by XBOOLE_1:4;
then Y in Z3 \/ (Z4 \/ Z5 \/ Z6) \/ Z7 by XBOOLE_1:4;
then A20: Y in Z3 \/ (Z4 \/ Z5 \/ Z6 \/ Z7) by XBOOLE_1:4;
now assume
A21:  Y in Z3;
then consider Y1,Y2,Y3,Y4 such that
A22:    Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y & Y1 meets X by A3;
Y in union union union X by A3,A21;
then Y4 in union union union union X by A22,TARSKI:def 4;
then Y4 in Z4 by A4,A22;
then Y4 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_0:def 2;
then Y4 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_0:def 2;
then Y4 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_0:def 2;
then Y4 in V by XBOOLE_0:def 2;
end;
then Y in Z4 \/ Z5 \/ Z6 \/ Z7 by A20,XBOOLE_0:def 2;
then Y in Z4 \/ (Z5 \/ Z6) \/ Z7 by XBOOLE_1:4;
then A23: Y in Z4 \/ (Z5 \/ Z6 \/ Z7) by XBOOLE_1:4;
now assume
A24:  Y in Z4;
then consider Y1,Y2,Y3 such that
A25:    Y1 in Y2 & Y2 in Y3 & Y3 in Y & Y1 meets X by A4;
Y in union union union union X by A4,A24;
then Y3 in union union union union union X by A25,TARSKI:def 4;
then Y3 in Z5 by A5,A25;
then Y3 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_0:def 2;
then Y3 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_0:def 2;
then Y3 in V by XBOOLE_0:def 2;
end;
then Y in Z5 \/ Z6 \/ Z7 by A23,XBOOLE_0:def 2;
then A26: Y in Z5 \/ (Z6 \/ Z7) by XBOOLE_1:4;
now assume
A27:  Y in Z5;
then consider Y1,Y2 such that
A28:    Y1 in Y2 & Y2 in Y & Y1 meets X by A5;
Y in union union union union union X by A5,A27;
then Y2 in union union union union union union X by A28,TARSKI:def 4;
then Y2 in Z6 by A6,A28;
then Y2 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_0:def 2;
then Y2 in V by XBOOLE_0:def 2;
end;
then A29: Y in Z6 \/ Z7 by A26,XBOOLE_0:def 2;
now assume
A30:  Y in Z6;
then consider Y1 such that
A31:    Y1 in Y & Y1 meets X by A6;
Y in union union union union union union X by A6,A30;
then Y1 in
union union union union union union union X by A31,TARSKI:def 4;
then Y1 in Z7 by A7,A31;
then Y1 in V by XBOOLE_0:def 2;
end;
then Y in Z7 by A29,XBOOLE_0:def 2;
then Y meets X by A7;
end;

::
::   Tuples for n=5
::

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

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

canceled;

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

theorem
Th6: [x1,x2,x3,x4,x5] = [[x1,x2],x3,x4,x5]
proof
thus [x1,x2,x3,x4,x5] = [[[[x1,x2],x3],x4],x5] by Th3
.= [[[x1,x2],x3],x4,x5] by MCART_1:def 3
.= [[x1,x2],x3,x4,x5] by MCART_1:32;
end;

theorem Th7:
[x1,x2,x3,x4,x5] = [y1,y2,y3,y4,y5]
implies x1 = y1 & x2 = y2 & x3 = y3 & x4 = y4 & x5 = y5
proof
assume [x1,x2,x3,x4,x5] = [y1,y2,y3,y4,y5];
then [[x1,x2,x3,x4],x5] = [y1,y2,y3,y4,y5] by Def1
.= [[y1,y2,y3,y4],y5] by Def1;
then [x1,x2,x3,x4] = [y1,y2,y3,y4]
& x5 = y5 by ZFMISC_1:33;
hence thesis by MCART_1:33;
end;

theorem Th8:
X <> {} implies
ex x st x in X &
not ex x1,x2,x3,x4,x5 st (x1 in X or x2 in X) & x = [x1,x2,x3,x4,x5]
proof
assume X <> {};
then consider Y such that
A1:  Y in X and
A2:  for 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 Y
holds Y1 misses X by Th2;
take x = Y;
thus x in X by A1;
given x1,x2,x3,x4,x5 such that
A3:  x1 in X or x2 in X and
A4:  x = [x1,x2,x3,x4,x5];
set Y1 = { x1, x2 },
Y2 = { Y1, {x1} },
Y3 = { Y2, x3 },
Y4 = { Y3, {Y2} },
Y5 = { Y4, x4 },
Y6 = { Y5, {Y4} },
Y7 = { Y6, x5 };
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] by A4,Th3
.= [[[ Y2,x3],x4],x5 ] by TARSKI:def 5
.= [[ Y4,x4],x5 ] by TARSKI:def 5
.= [ Y6,x5 ] by TARSKI:def 5
.= { Y7, { Y6 } } 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
Y
by TARSKI:def 2;
end;

::
::   Cartesian products of five sets
::

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

theorem Th9:
[:X1,X2,X3,X4,X5:] = [:[:[:[:X1,X2:],X3:],X4:],X5:]
proof
thus [:X1,X2,X3,X4,X5:] = [:[:X1,X2,X3,X4:],X5:] by Def2
.= [:[:[:X1,X2,X3:],X4:],X5:] by ZFMISC_1:def 4
.= [:[:[:[:X1,X2:],X3:],X4:],X5:] by ZFMISC_1:def 3;
end;

canceled;

theorem
[:X1,X2,X3,X4,X5:] = [:[:X1,X2,X3:],X4,X5:]
proof
thus [:X1,X2,X3,X4,X5:] = [:[:[:[:X1,X2:],X3:],X4:],X5:] by Th9
.= [:[:[:X1,X2:],X3:],X4,X5:] by ZFMISC_1:def 3
.= [:[:X1,X2,X3:],X4,X5:] by ZFMISC_1:def 3;
end;

theorem
Th12: [:X1,X2,X3,X4,X5:] = [:[:X1,X2:],X3,X4,X5:]
proof
thus [:X1,X2,X3,X4,X5:] = [:[:[:[:X1,X2:],X3:],X4:],X5:] by Th9
.= [:[:[:X1,X2:],X3:],X4,X5:] by ZFMISC_1:def 3
.= [:[:X1,X2:],X3,X4,X5:] by MCART_1:54;
end;

theorem
Th13: X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {}
iff [:X1,X2,X3,X4,X5:] <> {}
proof
A1: [:[:X1,X2,X3,X4:],X5:] = [:X1,X2,X3,X4,X5:] by Def2;
X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {}
iff [:X1,X2,X3,X4:] <> {} by MCART_1:55;
hence thesis by A1,ZFMISC_1:113;
end;

theorem Th14:
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} implies
( [:X1,X2,X3,X4,X5:] = [:Y1,Y2,Y3,Y4,Y5:]
implies X1=Y1 & X2=Y2 & X3=Y3 & X4=Y4 & X5=Y5 )
proof
A1: [:X1,X2,X3,X4,X5:] = [:[:X1,X2,X3,X4:],X5:] by Def2;
assume
A2: X1<>{} & X2<>{} & X3<>{} & X4<>{};
then A3: [:X1,X2,X3,X4:] <> {} by MCART_1:55;
assume
A4: X5<>{};
assume [:X1,X2,X3,X4,X5:] = [:Y1,Y2,Y3,Y4,Y5:];
then [:[:X1,X2,X3,X4:],X5:] = [:[:Y1,Y2,Y3,Y4:],Y5:] by A1,Def2;
then [:X1,X2,X3,X4:] = [:Y1,Y2,Y3,Y4:] & X5 = Y5 by A3,A4,ZFMISC_1:134;
hence thesis by A2,MCART_1:56;
end;

theorem
[:X1,X2,X3,X4,X5:]<>{} & [:X1,X2,X3,X4,X5:] = [:Y1,Y2,Y3,Y4,Y5:]
implies X1=Y1 & X2=Y2 & X3=Y3 & X4=Y4 & X5=Y5
proof
assume [:X1,X2,X3,X4,X5:]<>{};
then X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} by Th13;
hence thesis by Th14;
end;

theorem
[:X,X,X,X,X:] = [:Y,Y,Y,Y,Y:] implies X = Y
proof
assume [:X,X,X,X,X:] = [:Y,Y,Y,Y,Y:];
then X<>{} or Y<>{} implies thesis by Th14;
hence X = Y;
end;

theorem Th17:
X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} implies
for x being Element of [:X1,X2,X3,X4,X5:]
ex xx1,xx2,xx3,xx4,xx5 st x = [xx1,xx2,xx3,xx4,xx5]
proof
assume
A1:  X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {};
then A2:  [:X1,X2,X3,X4:] <> {} by MCART_1:55;
let x be Element of [:X1,X2,X3,X4,X5:];
reconsider x'=x as Element of [:[:X1,X2,X3,X4:],X5:] by Def2;
consider x1234 being (Element of [:X1,X2,X3,X4:]), xx5 such that
A3:   x' = [x1234,xx5] by A1,A2,Lm1;
consider xx1,xx2,xx3,xx4 such that
A4:   x1234 = [xx1,xx2,xx3,xx4] by A1,Lm3;
take xx1,xx2,xx3,xx4,xx5;
thus x = [xx1,xx2,xx3,xx4,xx5] by A3,A4,Def1;
end;

definition
let X1,X2,X3,X4,X5;
assume A1:X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{};
let x be Element of [:X1,X2,X3,X4,X5:];
func x`1 -> Element of X1 means
:Def3: x = [x1,x2,x3,x4,x5] implies it = x1;
existence
proof
consider xx1,xx2,xx3,xx4,xx5 such that
A2:   x = [xx1,xx2,xx3,xx4,xx5] by A1,Th17;
take xx1;
thus thesis by A2,Th7;
end;
uniqueness
proof
consider xx1,xx2,xx3,xx4,xx5 such that
A3:   x = [xx1,xx2,xx3,xx4,xx5] by A1,Th17;
let y,z be Element of X1;
assume x = [x1,x2,x3,x4,x5] implies y = x1;
then A4:   y = xx1 by A3;
assume x = [x1,x2,x3,x4,x5] implies z = x1;
hence thesis by A3,A4;
end;

func x`2 -> Element of X2 means
:Def4: x = [x1,x2,x3,x4,x5] implies it = x2;
existence
proof
consider xx1,xx2,xx3,xx4,xx5 such that
A5:   x = [xx1,xx2,xx3,xx4,xx5] by A1,Th17;
take xx2; thus thesis by A5,Th7;
end;
uniqueness
proof
consider xx1,xx2,xx3,xx4,xx5 such that
A6:   x = [xx1,xx2,xx3,xx4,xx5] by A1,Th17;
let y,z be Element of X2;
assume x = [x1,x2,x3,x4,x5] implies y = x2;
then A7:   y = xx2 by A6;
assume x = [x1,x2,x3,x4,x5] implies z = x2;
hence thesis by A6,A7;
end;

func x`3 -> Element of X3 means
:Def5: x = [x1,x2,x3,x4,x5] implies it = x3;
existence
proof
consider xx1,xx2,xx3,xx4,xx5 such that
A8:   x = [xx1,xx2,xx3,xx4,xx5] by A1,Th17;
take xx3; thus thesis by A8,Th7;
end;
uniqueness
proof
consider xx1,xx2,xx3,xx4,xx5 such that
A9:   x = [xx1,xx2,xx3,xx4,xx5] by A1,Th17;
let y,z be Element of X3;
assume x = [x1,x2,x3,x4,x5] implies y = x3;
then A10:   y = xx3 by A9;
assume x = [x1,x2,x3,x4,x5] implies z = x3;
hence thesis by A9,A10;
end;

func x`4 -> Element of X4 means
:Def6: x = [x1,x2,x3,x4,x5] implies it = x4;
existence
proof
consider xx1,xx2,xx3,xx4,xx5 such that
A11:   x = [xx1,xx2,xx3,xx4,xx5] by A1,Th17;
take xx4; thus thesis by A11,Th7;
end;
uniqueness
proof
consider xx1,xx2,xx3,xx4,xx5 such that
A12:   x = [xx1,xx2,xx3,xx4,xx5] by A1,Th17;
let y,z be Element of X4;
assume x = [x1,x2,x3,x4,x5] implies y = x4;
then A13:   y = xx4 by A12;
assume x = [x1,x2,x3,x4,x5] implies z = x4;
hence thesis by A12,A13;
end;

func x`5 -> Element of X5 means
:Def7: x = [x1,x2,x3,x4,x5] implies it = x5;
existence
proof
consider xx1,xx2,xx3,xx4,xx5 such that
A14:   x = [xx1,xx2,xx3,xx4,xx5] by A1,Th17;
take xx5; thus thesis by A14,Th7;
end;
uniqueness
proof
consider xx1,xx2,xx3,xx4,xx5 such that
A15:   x = [xx1,xx2,xx3,xx4,xx5] by A1,Th17;
let y,z be Element of X5;
assume x = [x1,x2,x3,x4,x5] implies y = x5;
then A16:   y = xx5 by A15;
assume x = [x1,x2,x3,x4,x5] implies z = x5;
hence thesis by A15,A16;
end;
end;

theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} implies
for x being Element of [:X1,X2,X3,X4,X5:]
for x1,x2,x3,x4,x5 st x = [x1,x2,x3,x4,x5] holds
x`1 = x1 & x`2 = x2 & x`3 = x3 & x`4 = x4 & x`5 = x5
by Def3,Def4,Def5,Def6,Def7;

theorem Th19:
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} implies
for x being Element of [:X1,X2,X3,X4,X5:] holds x = [x`1,x`2,x`3,x`4,x`5]
proof
assume
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{};
let x be Element of [:X1,X2,X3,X4,X5:];
consider xx1,xx2,xx3,xx4,xx5 such that
A2:  x = [xx1,xx2,xx3,xx4,xx5] by A1,Th17;
thus x = [x`1,xx2,xx3,xx4,xx5] by A1,A2,Def3
.= [x`1,x`2,xx3,xx4,xx5] by A1,A2,Def4
.= [x`1,x`2,x`3,xx4,xx5] by A1,A2,Def5
.= [x`1,x`2,x`3,x`4,xx5] by A1,A2,Def6
.= [x`1,x`2,x`3,x`4,x`5] by A1,A2,Def7;
end;

theorem
Th20:
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} implies
for x being Element of [:X1,X2,X3,X4,X5:] holds
x`1 = (x qua set)`1`1`1`1 &
x`2 = (x qua set)`1`1`1`2 &
x`3 = (x qua set)`1`1`2 &
x`4 = (x qua set)`1`2 &
x`5 = (x qua set)`2
proof
assume
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{};
let x be Element of [:X1,X2,X3,X4,X5:];
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 Def1
.= (x qua set)`1`1`1`1 by A1,Th19;
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 Def1
.= (x qua set)`1`1`1`2 by A1,Th19;
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 Def1
.= (x qua set)`1`1`2 by A1,Th19;
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 Def1
.= (x qua set)`1`2 by A1,Th19;
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 Def1
.= (x qua set)`2 by A1,Th19;
end;

theorem
X1 c= [:X1,X2,X3,X4,X5:]
or X1 c= [:X2,X3,X4,X5,X1:]
or X1 c= [:X3,X4,X5,X1,X2:]
or X1 c= [:X4,X5,X1,X2,X3:]
or X1 c= [:X5,X1,X2,X3,X4:]
implies X1 = {}
proof
assume that
A1:    X1 c= [:X1,X2,X3,X4,X5:]
or X1 c= [:X2,X3,X4,X5,X1:]
or X1 c= [:X3,X4,X5,X1,X2:]
or X1 c= [:X4,X5,X1,X2,X3:]
or X1 c= [:X5,X1,X2,X3,X4:]
and
A2: X1 <> {};
[:X1,X2,X3,X4,X5:]<>{}
or [:X2,X3,X4,X5,X1:]<>{}
or [:X3,X4,X5,X1,X2:]<>{}
or [:X4,X5,X1,X2,X3:]<>{}
or [:X5,X1,X2,X3,X4:]<>{} by A1,A2,XBOOLE_1:3;
then A3: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} by Th13;
now per cases by A1;
suppose
A4:  X1 c= [:X1,X2,X3,X4,X5:];
consider x such that
A5:   x in X1 and
A6:   for x1,x2,x3,x4,x5
st x1 in X1 or x2 in X1 holds x <> [x1,x2,x3,x4,x5] by A2,Th8;
reconsider x as Element of [:X1,X2,X3,X4,X5:] by A4,A5;
x = [x`1,x`2,x`3,x`4,x`5] & x`1 in X1 by A3,Th19;

suppose X1 c= [:X2,X3,X4,X5,X1:];
then X1 c= [:[:X2,X3:],X4,X5,X1:] by Th12;
hence thesis by A2,MCART_1:63;

suppose X1 c= [:X3,X4,X5,X1,X2:];
then X1 c= [:[:X3,X4:],X5,X1,X2:] by Th12;
hence thesis by A2,MCART_1:63;

suppose X1 c= [:X4,X5,X1,X2,X3:];
then X1 c= [:[:X4,X5:],X1,X2,X3:] by Th12;
hence thesis by A2,MCART_1:63;

suppose
A7:  X1 c= [:X5,X1,X2,X3,X4:];
consider x such that
A8:   x in X1 and
A9:   for x1,x2,x3,x4,x5
st x1 in X1 or x2 in X1 holds x <> [x1,x2,x3,x4,x5] by A2,Th8;
reconsider x as Element of [:X5,X1,X2,X3,X4:] by A7,A8;
x = [x`1,x`2,x`3,x`4,x`5] & x`2 in X1 by A3,Th19;
hence thesis by A9;
end;
end;

theorem
[:X1,X2,X3,X4,X5:] meets [:Y1,Y2,Y3,Y4,Y5:] implies
X1 meets Y1 & X2 meets Y2 & X3 meets Y3 & X4 meets Y4 & X5 meets Y5
proof
A1: [:[:[:[:X1,X2:],X3:],X4:],X5:] = [:X1,X2,X3,X4,X5:]
& [:[:[:[:Y1,Y2:],Y3:],Y4:],Y5:] = [:Y1,Y2,Y3,Y4,Y5:] by Th9;
assume [:X1,X2,X3,X4,X5:] meets [:Y1,Y2,Y3,Y4,Y5:];
then [:[:[:X1,X2:],X3:],X4:] meets [:[:[:Y1,Y2:],Y3:],Y4:]
& X5 meets Y5 by A1,ZFMISC_1:127;
then [:[:X1,X2:],X3:] meets [:[:Y1,Y2:],Y3:]
& X4 meets Y4 & X5 meets Y5 by ZFMISC_1:127;
then [:X1,X2:] meets [:Y1,Y2:]
& X3 meets Y3 & X4 meets Y4 & X5 meets Y5 by ZFMISC_1:127;
hence thesis by ZFMISC_1:127;
end;

theorem [:{x1},{x2},{x3},{x4},{x5}:] = { [x1,x2,x3,x4,x5] }
proof thus
[:{x1},{x2},{x3},{x4},{x5}:]
= [:[:{x1},{x2}:],{x3},{x4},{x5}:] by Th12
.= [:{[x1,x2]}, {x3},{x4},{x5}:] by ZFMISC_1:35
.= { [[x1,x2], x3, x4, x5]} by MCART_1:65
.= { [x1,x2,x3,x4,x5] } by Th6;
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;

:: 5 - Tuples

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

theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} implies
for x1,x2,x3,x4,x5 st x = [x1,x2,x3,x4,x5]
holds x`1 = x1 & x`2 = x2 & x`3 = x3 & x`4 = x4 & x`5 = x5 by Def3,Def4,Def5
,Def6,Def7;

theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} &
(for xx1,xx2,xx3,xx4,xx5 st x = [xx1,xx2,xx3,xx4,xx5] holds y1 = xx1)
implies y1 =x`1
proof
assume that
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} and
A2: for xx1,xx2,xx3,xx4,xx5 st x = [xx1,xx2,xx3,xx4,xx5] holds y1 = xx1;
x = [x`1,x`2,x`3,x`4,x`5] by A1,Th19;
hence thesis by A2;
end;

theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} &
(for xx1,xx2,xx3,xx4,xx5 st x = [xx1,xx2,xx3,xx4,xx5] holds y2 = xx2)
implies y2 =x`2
proof
assume that
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} and
A2: for xx1,xx2,xx3,xx4,xx5 st x = [xx1,xx2,xx3,xx4,xx5] holds y2 = xx2;
x = [x`1,x`2,x`3,x`4,x`5] by A1,Th19;
hence thesis by A2;
end;

theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} &
(for xx1,xx2,xx3,xx4,xx5 st x = [xx1,xx2,xx3,xx4,xx5] holds y3 = xx3)
implies y3 =x`3
proof
assume that
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} and
A2: for xx1,xx2,xx3,xx4,xx5 st x = [xx1,xx2,xx3,xx4,xx5] holds y3 = xx3;
x = [x`1,x`2,x`3,x`4,x`5] by A1,Th19;
hence thesis by A2;
end;

theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} &
(for xx1,xx2,xx3,xx4,xx5 st x = [xx1,xx2,xx3,xx4,xx5] holds y4 = xx4)
implies y4 =x`4
proof
assume that
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} and
A2: for xx1,xx2,xx3,xx4,xx5 st x = [xx1,xx2,xx3,xx4,xx5] holds y4 = xx4;
x = [x`1,x`2,x`3,x`4,x`5] by A1,Th19;
hence thesis by A2;
end;

theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} &
(for xx1,xx2,xx3,xx4,xx5 st x = [xx1,xx2,xx3,xx4,xx5] holds y5 = xx5)
implies y5 =x`5
proof
assume that
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} and
A2: for xx1,xx2,xx3,xx4,xx5 st x = [xx1,xx2,xx3,xx4,xx5] holds y5 = xx5;
x = [x`1,x`2,x`3,x`4,x`5] by A1,Th19;
hence thesis by A2;
end;

theorem
y in [: X1,X2,X3,X4,X5 :] implies
ex x1,x2,x3,x4,x5
st x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5
& y = [x1,x2,x3,x4,x5]
proof
assume y in [: X1,X2,X3,X4,X5 :];
then y in [:[:X1,X2,X3,X4:],X5:] by Def2;
then consider x1234, x5 being set such that
A1:   x1234 in [:X1,X2,X3,X4:] and
A2:   x5 in X5 and
A3:   y = [x1234,x5] by ZFMISC_1:def 2;
consider x1, x2, x3, x4 such that
A4:   x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4
& x1234 = [x1,x2,x3,x4] by A1,MCART_1:83;
y = [x1,x2,x3,x4,x5] by A3,A4,Def1;
hence thesis by A2,A4;
end;

theorem
[x1,x2,x3,x4,x5] in [: X1,X2,X3,X4,X5 :]
iff x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5
proof
A1: [:X1,X2,X3,X4,X5:] = [:[:X1,X2:],X3,X4,X5:] by Th12;
A2: [x1,x2,x3,x4,x5] = [[x1,x2],x3,x4,x5] by Th6;
[x1,x2] in [:X1,X2:] iff x1 in X1 & x2 in X2 by ZFMISC_1:106;
hence thesis by A1,A2,MCART_1:84;
end;

theorem
(for y holds y in Z iff
ex x1,x2,x3,x4,x5
st x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5
& y = [x1,x2,x3,x4,x5])
implies Z = [: X1,X2,X3,X4,X5 :]
proof
assume
A1:   for y holds y in Z iff
ex x1,x2,x3,x4,x5
st x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5
& y = [x1,x2,x3,x4,x5];
now let y;
thus y in Z implies y in [:[:X1,X2,X3,X4:],X5:]
proof
assume y in Z; then consider x1,x2,x3,x4,x5 such that
A2:      x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5
& y = [x1,x2,x3,x4,x5] by A1;
A3:      y = [[x1,x2,x3,x4],x5] by A2,Def1;
[x1,x2,x3,x4] in [:X1,X2,X3,X4:] & x5 in X5 by A2,MCART_1:84;
hence y in [:[:X1,X2,X3,X4:],X5:] by A3,ZFMISC_1:def 2;
end;
assume y in [:[:X1,X2,X3,X4:],X5:];
then consider x1234,x5 being set such that
A4:     x1234 in [:X1,X2,X3,X4:] and
A5:     x5 in X5 and
A6:     y = [x1234,x5] by ZFMISC_1:def 2;
consider x1,x2,x3,x4 such that
A7:   x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4
& x1234 = [x1,x2,x3,x4] by A4,MCART_1:83;
y = [x1,x2,x3,x4,x5] by A6,A7,Def1;
hence y in Z by A1,A5,A7;
end;
then Z = [:[:X1,X2,X3,X4:],X5:] by TARSKI:2;
hence Z = [: X1,X2,X3,X4,X5 :] by Def2;
end;

theorem Th33:
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{}
& Y1<>{} & Y2<>{} & Y3<>{} & Y4<>{} & Y5<>{} implies
for x being Element of [:X1,X2,X3,X4,X5:],
y being Element of [:Y1,Y2,Y3,Y4,Y5:]
holds x = y
implies x`1 = y`1 & x`2 = y`2 & x`3 = y`3 & x`4 = y`4 & x`5 = y`5
proof
assume that
A1: X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} and
A2: Y1 <> {} & Y2 <> {} & Y3 <> {} & Y4 <> {} & Y5 <> {};
let x be Element of [:X1,X2,X3,X4,X5:];
let y be Element of [:Y1,Y2,Y3,Y4,Y5:];
assume
A3: x = y;
thus x`1 = (x qua set)`1`1`1`1 by A1,Th20 .= y`1 by A2,A3,Th20;
thus x`2 = (x qua set)`1`1`1`2 by A1,Th20 .= y`2 by A2,A3,Th20;
thus x`3 = (x qua set)`1`1`2 by A1,Th20 .= y`3 by A2,A3,Th20;
thus x`4 = (x qua set)`1`2 by A1,Th20 .= y`4 by A2,A3,Th20;
thus x`5 = (x qua set)`2 by A1,Th20 .= y`5 by A2,A3,Th20;
end;

theorem
for x being Element of [:X1,X2,X3,X4,X5:] st x in [:A1,A2,A3,A4,A5:]
holds x`1 in A1 & x`2 in A2 & x`3 in A3 & x`4 in A4 & x`5 in A5
proof
let x be Element of [:X1,X2,X3,X4,X5:];
assume
A1: x in [:A1,A2,A3,A4,A5:];
then A2: A1 <> {} & A2 <> {} & A3 <> {} & A4 <> {} & A5 <> {} by Th13;
reconsider y = x as Element of [:A1,A2,A3,A4,A5:] by A1;
y`1 in A1 & y`2 in A2 & y`3 in A3 & y`4 in A4 & y`5 in A5 by A2;
hence x`1 in A1 & x`2 in A2 & x`3 in A3 & x`4 in A4 & x`5 in A5 by Th33;
end;

theorem Th35:
X1 c= Y1 & X2 c= Y2 & X3 c= Y3 & X4 c= Y4 & X5 c= Y5
implies [:X1,X2,X3,X4,X5:] c= [:Y1,Y2,Y3,Y4,Y5:]
proof
assume X1 c= Y1 & X2 c= Y2 & X3 c= Y3 & X4 c= Y4;
then A1: [:X1,X2,X3,X4:] c= [:Y1,Y2,Y3,Y4:] by MCART_1:88;
A2: [:X1,X2,X3,X4,X5:] = [:[:X1,X2,X3,X4:],X5:] &
[:Y1,Y2,Y3,Y4,Y5:] = [:[:Y1,Y2,Y3,Y4:],Y5:] by Def2;
assume X5 c= Y5;
hence thesis by A1,A2,ZFMISC_1:119;
end;

definition
let X1,X2,X3,X4,X5,A1,A2,A3,A4,A5;
redefine func [:A1,A2,A3,A4,A5:] -> Subset of [:X1,X2,X3,X4,X5:];
coherence by Th35;
end;

theorem
X1 <> {} & X2 <> {} implies
for xx being Element of [:X1,X2:]
ex xx1 being (Element of X1),
xx2 being Element of X2 st xx = [xx1,xx2] by Lm1;

theorem
X1 <> {} & X2 <> {} & X3 <> {} implies
for xx being Element of [:X1,X2,X3:]
ex xx1,xx2,xx3 st xx = [xx1,xx2,xx3] by Lm2;

theorem
X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} implies
for xx being Element of [:X1,X2,X3,X4:]
ex xx1,xx2,xx3,xx4 st xx = [xx1,xx2,xx3,xx4] by Lm3;

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