### The Mizar article:

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

**by****Michal Muzalewski, and****Wojciech Skaba**

- Received October 15, 1990
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; hence contradiction by A13,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; hence contradiction by A13,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; hence contradiction by A13,A22,XBOOLE_0:3; 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; hence contradiction by A13,A25,XBOOLE_0:3; 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; hence contradiction by A13,A28,XBOOLE_0:3; 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; hence contradiction by A13,A31,XBOOLE_0:3; 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; hence contradiction by A13,A34,XBOOLE_0:3; 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; hence contradiction by A13,A37,XBOOLE_0:3; 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; hence contradiction by A13,A40,XBOOLE_0:3; 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; hence contradiction by A13,A43,XBOOLE_0:3; end; then Y in ZA by A41,XBOOLE_0:def 2; then Y meets X by A10; hence contradiction by A11,A13,XBOOLE_1:70; 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; hence contradiction by A14,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; hence contradiction by A14,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; hence contradiction by A14,A22,XBOOLE_0:3; 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; hence contradiction by A14,A24,XBOOLE_0:3; 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; hence contradiction by A14,A26,XBOOLE_0:3; 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; hence contradiction by A14,A28,XBOOLE_0:3; 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; hence contradiction by A14,A30,XBOOLE_0:3; 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; hence contradiction by A14,A32,XBOOLE_0:3; 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; hence contradiction by A14,A34,XBOOLE_0:3; 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; hence contradiction by A14,A36,XBOOLE_0:3; 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; hence contradiction by A14,A38,XBOOLE_0:3; suppose Y in ZB; then Y meets X by A11; hence contradiction by A12,A14,XBOOLE_1:70; 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; hence contradiction by A2,A5; 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; hence contradiction by A6; 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; hence contradiction; 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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