let X be set ; :: thesis: ( X <> {} implies ex Y being set st
( Y in X & ( for Y1, Y2, Y3, Y4, Y5, Y6, Y7, Y8, Y9, YA, YB, YC, YD, YE being set st Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in Y9 & Y9 in YA & YA in YB & YB in YC & YC in YD & YD in YE & YE in Y holds
Y1 misses X ) ) )
defpred S1[ set ] means ex Y1, Y2, Y3, Y4, Y5, Y6, Y7, Y8, Y9, YA, YB, YC, YD being set st
( Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in Y9 & Y9 in YA & YA in YB & YB in YC & YC in YD & YD in $1 & Y1 meets X );
consider Z1 being set such that
A1: for Y being set holds
( Y in Z1 iff ( Y in union X & S1[Y] ) ) from XBOOLE_0:sch 1();
 defpred S2[ set ] means $1 meets X;
defpred S3[ set ] means ex Y1 being set st
( Y1 in $1 & Y1 meets X );
defpred S4[ set ] means ex Y1, Y2 being set st
( Y1 in Y2 & Y2 in $1 & Y1 meets X );
defpred S5[ set ] means ex Y1, Y2, Y3 being set st
( Y1 in Y2 & Y2 in Y3 & Y3 in $1 & Y1 meets X );
defpred S6[ set ] means ex Y1, Y2, Y3, Y4 being set st
( Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in $1 & Y1 meets X );
defpred S7[ set ] means ex Y1, Y2, Y3, Y4, Y5 being set st
( Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in $1 & Y1 meets X );
defpred S8[ set ] means ex Y1, Y2, Y3, Y4, Y5, Y6 being set st
( Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in $1 & Y1 meets X );
defpred S9[ set ] means ex Y1, Y2, Y3, Y4, Y5, Y6, Y7 being set 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 );
defpred S10[ set ] means ex Y1, Y2, Y3, Y4, Y5, Y6, Y7, Y8 being set st
( Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in $1 & Y1 meets X );
defpred S11[ set ] means ex Y1, Y2, Y3, Y4, Y5, Y6, Y7, Y8, Y9 being set st
( Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in Y9 & Y9 in $1 & Y1 meets X );
defpred S12[ set ] means ex Y1, Y2, Y3, Y4, Y5, Y6, Y7, Y8, Y9, YA being set st
( Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in Y9 & Y9 in YA & YA in $1 & Y1 meets X );
defpred S13[ set ] means ex Y1, Y2, Y3, Y4, Y5, Y6, Y7, Y8, Y9, YA, YB being set st
( Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in Y9 & Y9 in YA & YA in YB & YB in $1 & Y1 meets X );
defpred S14[ set ] means ex Y1, Y2, Y3, Y4, Y5, Y6, Y7, Y8, Y9, YA, YB, YC being set st
( Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in Y9 & Y9 in YA & YA in YB & YB in YC & YC in $1 & Y1 meets X );
consider Z2 being set such that
A2: for Y being set holds
( Y in Z2 iff ( Y in union (union X) & S14[Y] ) ) from XBOOLE_0:sch 1();
 consider Z7 being set such that
A3: for Y being set holds
( Y in Z7 iff ( Y in union (union (union (union (union (union (union X)))))) & S9[Y] ) ) from XBOOLE_0:sch 1();
 consider Z6 being set such that
A4: for Y being set holds
( Y in Z6 iff ( Y in union (union (union (union (union (union X))))) & S10[Y] ) ) from XBOOLE_0:sch 1();
 consider ZE being set such that
A5: for Y being set holds
( Y in ZE iff ( Y in union (union (union (union (union (union (union (union (union (union (union (union (union (union X))))))))))))) & S2[Y] ) ) from XBOOLE_0:sch 1();
 consider Z3 being set such that
A6: for Y being set holds
( Y in Z3 iff ( Y in union (union (union X)) & S13[Y] ) ) from XBOOLE_0:sch 1();
 consider Z5 being set such that
A7: for Y being set holds
( Y in Z5 iff ( Y in union (union (union (union (union X)))) & S11[Y] ) ) from XBOOLE_0:sch 1();
 consider Z4 being set such that
A8: for Y being set holds
( Y in Z4 iff ( Y in union (union (union (union X))) & S12[Y] ) ) from XBOOLE_0:sch 1();
 consider ZD being set such that
A9: for Y being set holds
( Y in ZD iff ( Y in union (union (union (union (union (union (union (union (union (union (union (union (union X)))))))))))) & S3[Y] ) ) from XBOOLE_0:sch 1();
 consider ZC being set such that
A10: for Y being set holds
( Y in ZC iff ( Y in union (union (union (union (union (union (union (union (union (union (union (union X))))))))))) & S4[Y] ) ) from XBOOLE_0:sch 1();
 consider ZB being set such that
A11: for Y being set holds
( Y in ZB iff ( Y in union (union (union (union (union (union (union (union (union (union (union X)))))))))) & S5[Y] ) ) from XBOOLE_0:sch 1();
 consider ZA being set such that
A12: for Y being set holds
( Y in ZA iff ( Y in union (union (union (union (union (union (union (union (union (union X))))))))) & S6[Y] ) ) from XBOOLE_0:sch 1();
 consider Z9 being set such that
A13: for Y being set holds
( Y in Z9 iff ( Y in union (union (union (union (union (union (union (union (union X)))))))) & S7[Y] ) ) from XBOOLE_0:sch 1();
 consider Z8 being set such that
A14: for Y being set holds
( Y in Z8 iff ( Y in union (union (union (union (union (union (union (union X))))))) & S8[Y] ) ) from XBOOLE_0:sch 1();
 set V = (((((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD) \/ ZE;
A15: (((((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD) \/ ZE = ((((((((((((X \/ (Z1 \/ Z2)) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD) \/ ZE by XBOOLE_1:4
.= (((((((((((X \/ ((Z1 \/ Z2) \/ Z3)) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD) \/ ZE by XBOOLE_1:4
.= ((((((((((X \/ (((Z1 \/ Z2) \/ Z3) \/ Z4)) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD) \/ ZE by XBOOLE_1:4
.= (((((((((X \/ ((((Z1 \/ Z2) \/ Z3) \/ Z4) \/ Z5)) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD) \/ ZE by XBOOLE_1:4
.= ((((((((X \/ (((((Z1 \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6)) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD) \/ ZE by XBOOLE_1:4
.= (((((((X \/ ((((((Z1 \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7)) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD) \/ ZE by XBOOLE_1:4
.= ((((((X \/ (((((((Z1 \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8)) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD) \/ ZE by XBOOLE_1:4
.= (((((X \/ ((((((((Z1 \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9)) \/ ZA) \/ ZB) \/ ZC) \/ ZD) \/ ZE by XBOOLE_1:4
.= ((((X \/ (((((((((Z1 \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA)) \/ ZB) \/ ZC) \/ ZD) \/ ZE by XBOOLE_1:4
.= (((X \/ ((((((((((Z1 \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB)) \/ ZC) \/ ZD) \/ ZE by XBOOLE_1:4
.= ((X \/ (((((((((((Z1 \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC)) \/ ZD) \/ ZE by XBOOLE_1:4
.= (X \/ ((((((((((((Z1 \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD)) \/ ZE by XBOOLE_1:4
.= X \/ (((((((((((((Z1 \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD) \/ ZE) by XBOOLE_1:4 ;
assume X <> {} ; :: thesis: ex Y being set st
( Y in X & ( for Y1, Y2, Y3, Y4, Y5, Y6, Y7, Y8, Y9, YA, YB, YC, YD, YE being set st Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in Y9 & Y9 in YA & YA in YB & YB in YC & YC in YD & YD in YE & YE in Y holds
Y1 misses X ) )
then consider Y being set such that
A16: Y in (((((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD) \/ ZE and
A17: Y misses (((((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD) \/ ZE by MCART_1:1;
( Y in ((((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD or Y in ZE ) by A16, XBOOLE_0:def 3;
then ( Y in (((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC or Y in ZD or Y in ZE ) by XBOOLE_0:def 3;
then ( Y in ((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB or Y in ZC or Y in ZD or Y in ZE ) by XBOOLE_0:def 3;
then ( Y in (((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA or Y in ZB or Y in ZC or Y in ZD or Y in ZE ) by XBOOLE_0:def 3;
then ( Y in ((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9 or Y in ZA or Y in ZB or Y in ZC or Y in ZD or Y in ZE ) by XBOOLE_0:def 3;
then ( Y in (((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8 or Y in Z9 or Y in ZA or Y in ZB or Y in ZC or Y in ZD or Y in ZE ) by XBOOLE_0:def 3;
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 or Y in ZC or Y in ZD or Y in ZE ) by XBOOLE_0:def 3;
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 or Y in ZC or Y in ZD or Y in ZE ) by XBOOLE_0:def 3;
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 or Y in ZC or Y in ZD or Y in ZE ) by XBOOLE_0:def 3;
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 or Y in ZC or Y in ZD or Y in ZE ) by XBOOLE_0:def 3;
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 or Y in ZC or Y in ZD or Y in ZE ) by XBOOLE_0:def 3;
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 or Y in ZC or Y in ZD or Y in ZE ) by XBOOLE_0:def 3;
then A18: ( 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 or Y in ZC or Y in ZD or Y in ZE ) by XBOOLE_0:def 3;
assume A19: for Y being set holds
( not Y in X or ex Y1, Y2, Y3, Y4, Y5, Y6, Y7, Y8, Y9, YA, YB, YC, YD, YE being set st
( Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in Y9 & Y9 in YA & YA in YB & YB in YC & YC in YD & YD in YE & YE in Y & not Y1 misses X ) ) ; :: thesis: contradiction
per cases ( Y in X or Y in 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 or Y in ZC or Y in ZD or Y in ZE ) by A18, XBOOLE_0:def 3;
suppose A20: Y in X ; :: thesis: contradiction
set Z15 = ((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5;
consider Y1, Y2, Y3, Y4, Y5, Y6, Y7, Y8, Y9, YA, YB, YC, YD, YE being set such that
A21: ( Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in Y9 & Y9 in YA & YA in YB & YB in YC & YC in YD & YD in YE ) and
A22: YE in Y and
A23: not Y1 misses X by A19, A20;
YE in union X by A20, A22, TARSKI:def 4;
then YE in Z1 by A1, A21, A23;
then YE in X \/ Z1 by XBOOLE_0:def 3;
then YE in (X \/ Z1) \/ Z2 by XBOOLE_0:def 3;
then YE in ((X \/ Z1) \/ Z2) \/ Z3 by XBOOLE_0:def 3;
then YE in (((X \/ Z1) \/ Z2) \/ Z3) \/ Z4 by XBOOLE_0:def 3;
then YE in ((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5 by XBOOLE_0:def 3;
then Y meets ((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5 by A22, 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;
then Y meets ((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB by XBOOLE_1:70;
then Y meets (((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC by XBOOLE_1:70;
then Y meets ((((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD by XBOOLE_1:70;
hence contradiction by A17, XBOOLE_1:70; :: thesis: verum
end;
suppose A24: Y in Z1 ; :: thesis: contradiction
set Z15 = ((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5;
consider Y1, Y2, Y3, Y4, Y5, Y6, Y7, Y8, Y9, YA, YB, YC, YD being set such that
A25: ( Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in Y9 & Y9 in YA & YA in YB & YB in YC & YC in YD ) and
A26: YD in Y and
A27: Y1 meets X by A1, A24;
Y in union X by A1, A24;
then YD in union (union X) by A26, TARSKI:def 4;
then YD in Z2 by A2, A25, A27;
then YD in (X \/ Z1) \/ Z2 by XBOOLE_0:def 3;
then Y meets (X \/ Z1) \/ Z2 by A26, 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;
then Y meets ((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB by XBOOLE_1:70;
then Y meets (((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC by XBOOLE_1:70;
then Y meets ((((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD by XBOOLE_1:70;
hence contradiction by A17, XBOOLE_1:70; :: thesis: verum
end;
suppose A28: Y in Z2 ; :: thesis: contradiction
then consider Y1, Y2, Y3, Y4, Y5, Y6, Y7, Y8, Y9, YA, YB, YC being set such that
A29: ( Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in Y9 & Y9 in YA & YA in YB & YB in YC ) and
A30: YC in Y and
A31: Y1 meets X by A2;
Y in union (union X) by A2, A28;
then YC in union (union (union X)) by A30, TARSKI:def 4;
then YC in Z3 by A6, A29, A31;
then YC in ((X \/ Z1) \/ Z2) \/ Z3 by XBOOLE_0:def 3;
then YC in (((X \/ Z1) \/ Z2) \/ Z3) \/ Z4 by XBOOLE_0:def 3;
then YC in ((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5 by XBOOLE_0:def 3;
then YC in (((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6 by XBOOLE_0:def 3;
then YC in ((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7 by XBOOLE_0:def 3;
then YC in (((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8 by XBOOLE_0:def 3;
then YC in ((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9 by XBOOLE_0:def 3;
then YC in (((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA by XBOOLE_0:def 3;
then YC in ((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB by XBOOLE_0:def 3;
then YC in (((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC by XBOOLE_0:def 3;
then YC in ((((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD by XBOOLE_0:def 3;
then YC in (((((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD) \/ ZE by XBOOLE_0:def 3;
hence contradiction by A17, A30, XBOOLE_0:3; :: thesis: verum
end;
suppose A32: Y in Z3 ; :: thesis: contradiction
then consider Y1, Y2, Y3, Y4, Y5, Y6, Y7, Y8, Y9, YA, YB being set such that
A33: ( 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 ) and
A34: YB in Y and
A35: Y1 meets X by A6;
Y in union (union (union X)) by A6, A32;
then YB in union (union (union (union X))) by A34, TARSKI:def 4;
then YB in Z4 by A8, A33, A35;
then YB in (((X \/ Z1) \/ Z2) \/ Z3) \/ Z4 by XBOOLE_0:def 3;
then YB in ((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5 by XBOOLE_0:def 3;
then YB in (((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6 by XBOOLE_0:def 3;
then YB in ((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7 by XBOOLE_0:def 3;
then YB in (((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8 by XBOOLE_0:def 3;
then YB in ((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9 by XBOOLE_0:def 3;
then YB in (((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA by XBOOLE_0:def 3;
then YB in ((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB by XBOOLE_0:def 3;
then YB in (((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC by XBOOLE_0:def 3;
then YB in ((((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD by XBOOLE_0:def 3;
then YB in (((((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD) \/ ZE by XBOOLE_0:def 3;
hence contradiction by A17, A34, XBOOLE_0:3; :: thesis: verum
end;
suppose A36: Y in Z4 ; :: thesis: contradiction
then consider Y1, Y2, Y3, Y4, Y5, Y6, Y7, Y8, Y9, YA being set such that
A37: ( 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 ) and
A38: YA in Y and
A39: Y1 meets X by A8;
Y in union (union (union (union X))) by A8, A36;
then YA in union (union (union (union (union X)))) by A38, TARSKI:def 4;
then YA in Z5 by A7, A37, A39;
then YA in ((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5 by XBOOLE_0:def 3;
then YA in (((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6 by XBOOLE_0:def 3;
then YA in ((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7 by XBOOLE_0:def 3;
then YA in (((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8 by XBOOLE_0:def 3;
then YA in ((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9 by XBOOLE_0:def 3;
then YA in (((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA by XBOOLE_0:def 3;
then YA in ((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB by XBOOLE_0:def 3;
then YA in (((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC by XBOOLE_0:def 3;
then YA in ((((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD by XBOOLE_0:def 3;
then YA in (((((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD) \/ ZE by XBOOLE_0:def 3;
hence contradiction by A17, A38, XBOOLE_0:3; :: thesis: verum
end;
suppose A40: Y in Z5 ; :: thesis: contradiction
then consider Y1, Y2, Y3, Y4, Y5, Y6, Y7, Y8, Y9 being set such that
A41: ( Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in Y9 ) and
A42: Y9 in Y and
A43: Y1 meets X by A7;
Y in union (union (union (union (union X)))) by A7, A40;
then Y9 in union (union (union (union (union (union X))))) by A42, TARSKI:def 4;
then Y9 in Z6 by A4, A41, A43;
then Y9 in (((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6 by XBOOLE_0:def 3;
then Y9 in ((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7 by XBOOLE_0:def 3;
then Y9 in (((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8 by XBOOLE_0:def 3;
then Y9 in ((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9 by XBOOLE_0:def 3;
then Y9 in (((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA by XBOOLE_0:def 3;
then Y9 in ((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB by XBOOLE_0:def 3;
then Y9 in (((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC by XBOOLE_0:def 3;
then Y9 in ((((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD by XBOOLE_0:def 3;
then Y9 in (((((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD) \/ ZE by XBOOLE_0:def 3;
hence contradiction by A17, A42, XBOOLE_0:3; :: thesis: verum
end;
suppose A44: Y in Z6 ; :: thesis: contradiction
then consider Y1, Y2, Y3, Y4, Y5, Y6, Y7, Y8 being set such that
A45: ( Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 ) and
A46: Y8 in Y and
A47: Y1 meets X by A4;
Y in union (union (union (union (union (union X))))) by A4, A44;
then Y8 in union (union (union (union (union (union (union X)))))) by A46, TARSKI:def 4;
then Y8 in Z7 by A3, A45, A47;
then Y8 in ((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7 by XBOOLE_0:def 3;
then Y8 in (((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8 by XBOOLE_0:def 3;
then Y8 in ((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9 by XBOOLE_0:def 3;
then Y8 in (((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA by XBOOLE_0:def 3;
then Y8 in ((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB by XBOOLE_0:def 3;
then Y8 in (((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC by XBOOLE_0:def 3;
then Y8 in ((((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD by XBOOLE_0:def 3;
then Y8 in (((((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD) \/ ZE by XBOOLE_0:def 3;
hence contradiction by A17, A46, XBOOLE_0:3; :: thesis: verum
end;
suppose A48: Y in Z7 ; :: thesis: contradiction
then consider Y1, Y2, Y3, Y4, Y5, Y6, Y7 being set such that
A49: ( Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 ) and
A50: Y7 in Y and
A51: Y1 meets X by A3;
Y in union (union (union (union (union (union (union X)))))) by A3, A48;
then Y7 in union (union (union (union (union (union (union (union X))))))) by A50, TARSKI:def 4;
then Y7 in Z8 by A14, A49, A51;
then Y7 in (((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8 by XBOOLE_0:def 3;
then Y7 in ((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9 by XBOOLE_0:def 3;
then Y7 in (((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA by XBOOLE_0:def 3;
then Y7 in ((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB by XBOOLE_0:def 3;
then Y7 in (((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC by XBOOLE_0:def 3;
then Y7 in ((((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD by XBOOLE_0:def 3;
then Y7 in (((((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD) \/ ZE by XBOOLE_0:def 3;
hence contradiction by A17, A50, XBOOLE_0:3; :: thesis: verum
end;
suppose A52: Y in Z8 ; :: thesis: contradiction
then consider Y1, Y2, Y3, Y4, Y5, Y6 being set such that
A53: ( Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 ) and
A54: Y6 in Y and
A55: Y1 meets X by A14;
Y in union (union (union (union (union (union (union (union X))))))) by A14, A52;
then Y6 in union (union (union (union (union (union (union (union (union X)))))))) by A54, TARSKI:def 4;
then Y6 in Z9 by A13, A53, A55;
then Y6 in ((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9 by XBOOLE_0:def 3;
then Y6 in (((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA by XBOOLE_0:def 3;
then Y6 in ((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB by XBOOLE_0:def 3;
then Y6 in (((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC by XBOOLE_0:def 3;
then Y6 in ((((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD by XBOOLE_0:def 3;
then Y6 in (((((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD) \/ ZE by XBOOLE_0:def 3;
hence contradiction by A17, A54, XBOOLE_0:3; :: thesis: verum
end;
suppose A56: Y in Z9 ; :: thesis: contradiction
then consider Y1, Y2, Y3, Y4, Y5 being set such that
A57: ( Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 ) and
A58: Y5 in Y and
A59: Y1 meets X by A13;
Y in union (union (union (union (union (union (union (union (union X)))))))) by A13, A56;
then Y5 in union (union (union (union (union (union (union (union (union (union X))))))))) by A58, TARSKI:def 4;
then Y5 in ZA by A12, A57, A59;
then Y5 in (((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA by XBOOLE_0:def 3;
then Y5 in ((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB by XBOOLE_0:def 3;
then Y5 in (((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC by XBOOLE_0:def 3;
then Y5 in ((((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD by XBOOLE_0:def 3;
then Y5 in (((((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD) \/ ZE by XBOOLE_0:def 3;
hence contradiction by A17, A58, XBOOLE_0:3; :: thesis: verum
end;
suppose A60: Y in ZA ; :: thesis: contradiction
then consider Y1, Y2, Y3, Y4 being set such that
A61: ( Y1 in Y2 & Y2 in Y3 & Y3 in Y4 ) and
A62: Y4 in Y and
A63: Y1 meets X by A12;
Y in union (union (union (union (union (union (union (union (union (union X))))))))) by A12, A60;
then Y4 in union (union (union (union (union (union (union (union (union (union (union X)))))))))) by A62, TARSKI:def 4;
then Y4 in ZB by A11, A61, A63;
then Y4 in ((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB by XBOOLE_0:def 3;
then Y4 in (((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC by XBOOLE_0:def 3;
then Y4 in ((((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD by XBOOLE_0:def 3;
then Y4 in (((((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD) \/ ZE by XBOOLE_0:def 3;
hence contradiction by A17, A62, XBOOLE_0:3; :: thesis: verum
end;
suppose A64: Y in ZB ; :: thesis: contradiction
then consider Y1, Y2, Y3 being set such that
A65: ( Y1 in Y2 & Y2 in Y3 ) and
A66: Y3 in Y and
A67: Y1 meets X by A11;
Y in union (union (union (union (union (union (union (union (union (union (union X)))))))))) by A11, A64;
then Y3 in union (union (union (union (union (union (union (union (union (union (union (union X))))))))))) by A66, TARSKI:def 4;
then Y3 in ZC by A10, A65, A67;
then Y3 in (((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC by XBOOLE_0:def 3;
then Y3 in ((((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD by XBOOLE_0:def 3;
then Y3 in (((((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD) \/ ZE by XBOOLE_0:def 3;
hence contradiction by A17, A66, XBOOLE_0:3; :: thesis: verum
end;
suppose A68: Y in ZC ; :: thesis: contradiction
then consider Y1, Y2 being set such that
A69: Y1 in Y2 and
A70: Y2 in Y and
A71: Y1 meets X by A10;
Y in union (union (union (union (union (union (union (union (union (union (union (union X))))))))))) by A10, A68;
then Y2 in union (union (union (union (union (union (union (union (union (union (union (union (union X)))))))))))) by A70, TARSKI:def 4;
then Y2 in ZD by A9, A69, A71;
then Y2 in ((((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD by XBOOLE_0:def 3;
then Y2 in (((((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD) \/ ZE by XBOOLE_0:def 3;
hence contradiction by A17, A70, XBOOLE_0:3; :: thesis: verum
end;
suppose A72: Y in ZD ; :: thesis: contradiction
then consider Y1 being set such that
A73: Y1 in Y and
A74: Y1 meets X by A9;
Y in union (union (union (union (union (union (union (union (union (union (union (union (union X)))))))))))) by A9, A72;
then Y1 in union (union (union (union (union (union (union (union (union (union (union (union (union (union X))))))))))))) by A73, TARSKI:def 4;
then Y1 in ZE by A5, A74;
then Y1 in (((((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD) \/ ZE by XBOOLE_0:def 3;
hence contradiction by A17, A73, XBOOLE_0:3; :: thesis: verum
end;
suppose Y in ZE ; :: thesis: contradiction
then Y meets X by A5;
hence contradiction by A15, A17, XBOOLE_1:70; :: thesis: verum
end;
end;