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 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 Y holds
Y1 misses X ) ) )
defpred S1[ 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 );
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 );
consider Z2 being set such that
A2: for Y being set holds
( Y in Z2 iff ( Y in union (union X) & S9[Y] ) ) from XBOOLE_0:sch 1();
 consider Z8 being set such that
A3: for Y being set holds
( Y in Z8 iff ( Y in union (union (union (union (union (union (union (union X))))))) & S3[Y] ) ) from XBOOLE_0:sch 1();
 consider Z5 being set such that
A4: for Y being set holds
( Y in Z5 iff ( Y in union (union (union (union (union X)))) & S6[Y] ) ) from XBOOLE_0:sch 1();
 consider Z4 being set such that
A5: for Y being set holds
( Y in Z4 iff ( Y in union (union (union (union X))) & S7[Y] ) ) from XBOOLE_0:sch 1();
 consider Z9 being set such that
A6: for Y being set holds
( Y in Z9 iff ( Y in union (union (union (union (union (union (union (union (union X)))))))) & S2[Y] ) ) from XBOOLE_0:sch 1();
 consider Z3 being set such that
A7: for Y being set holds
( Y in Z3 iff ( Y in union (union (union X)) & S8[Y] ) ) from XBOOLE_0:sch 1();
 consider Z7 being set such that
A8: for Y being set holds
( Y in Z7 iff ( Y in union (union (union (union (union (union (union X)))))) & S4[Y] ) ) from XBOOLE_0:sch 1();
 consider Z6 being set such that
A9: for Y being set holds
( Y in Z6 iff ( Y in union (union (union (union (union (union X))))) & S5[Y] ) ) from XBOOLE_0:sch 1();
 set V = ((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9;
assume X <> {} ; :: thesis: ex Y being set st
( Y in X & ( for 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 Y holds
Y1 misses X ) )
then consider Y being set such that
A10: Y in ((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9 and
A11: Y misses ((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9 by MCART_1:1;
A12: ((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9 = (((((((X \/ (Z1 \/ Z2)) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9 by XBOOLE_1:4
.= ((((((X \/ ((Z1 \/ Z2) \/ Z3)) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9 by XBOOLE_1:4
.= (((((X \/ (((Z1 \/ Z2) \/ Z3) \/ Z4)) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9 by XBOOLE_1:4
.= ((((X \/ ((((Z1 \/ Z2) \/ Z3) \/ Z4) \/ Z5)) \/ Z6) \/ Z7) \/ Z8) \/ Z9 by XBOOLE_1:4
.= (((X \/ (((((Z1 \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6)) \/ Z7) \/ Z8) \/ Z9 by XBOOLE_1:4
.= ((X \/ ((((((Z1 \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7)) \/ Z8) \/ Z9 by XBOOLE_1:4
.= (X \/ (((((((Z1 \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8)) \/ Z9 by XBOOLE_1:4
.= X \/ ((((((((Z1 \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) by XBOOLE_1:4 ;
A13: now
assume A14: Y in Z1 ; :: thesis: contradiction
then consider Y1, Y2, Y3, Y4, Y5, Y6, Y7, Y8 being set such that
A15: ( Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 ) and
A16: Y8 in Y and
A17: Y1 meets X by A1;
Y in union X by A1, A14;
then Y8 in union (union X) by A16, TARSKI:def 4;
then Y8 in Z2 by A2, A15, A17;
then Y8 in (X \/ Z1) \/ Z2 by XBOOLE_0:def 3;
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;
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;
hence contradiction by A11, XBOOLE_1:70; :: thesis: verum
end;
A18: now
assume A19: Y in Z2 ; :: thesis: contradiction
then consider Y1, Y2, Y3, Y4, Y5, Y6, Y7 being set such that
A20: ( Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 ) and
A21: Y7 in Y and
A22: Y1 meets X by A2;
Y in union (union X) by A2, A19;
then Y7 in union (union (union X)) by A21, TARSKI:def 4;
then Y7 in Z3 by A7, A20, A22;
then Y7 in ((X \/ Z1) \/ Z2) \/ Z3 by XBOOLE_0:def 3;
then Y7 in (((X \/ Z1) \/ Z2) \/ Z3) \/ Z4 by XBOOLE_0:def 3;
then Y7 in ((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5 by XBOOLE_0:def 3;
then Y7 in (((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6 by XBOOLE_0:def 3;
then Y7 in ((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7 by XBOOLE_0:def 3;
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;
hence contradiction by A11, A21, XBOOLE_0:3; :: thesis: verum
end;
assume A23: for Y being set holds
( not Y in X or 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 Y & not Y1 misses X ) ) ; :: thesis: contradiction
now
assume A24: Y in X ; :: thesis: contradiction
then consider Y1, Y2, Y3, Y4, Y5, Y6, Y7, Y8, Y9 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 ) and
A26: Y9 in Y and
A27: not Y1 misses X by A23;
Y9 in union X by A24, A26, TARSKI:def 4;
then Y9 in Z1 by A1, A25, A27;
then Y9 in X \/ Z1 by XBOOLE_0:def 3;
then Y9 in (X \/ Z1) \/ Z2 by XBOOLE_0:def 3;
then Y9 in ((X \/ Z1) \/ Z2) \/ Z3 by XBOOLE_0:def 3;
then Y9 in (((X \/ Z1) \/ Z2) \/ Z3) \/ Z4 by XBOOLE_0:def 3;
then Y9 in ((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5 by XBOOLE_0:def 3;
then Y meets ((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5 by A26, 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;
hence contradiction by A11, XBOOLE_1:70; :: thesis: verum
end;
then Y in (((((((Z1 \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9 by A12, A10, XBOOLE_0:def 3;
then Y in ((((((Z1 \/ (Z2 \/ Z3)) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9 by XBOOLE_1:4;
then Y in (((((Z1 \/ ((Z2 \/ Z3) \/ Z4)) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9 by XBOOLE_1:4;
then Y in ((((Z1 \/ (((Z2 \/ Z3) \/ Z4) \/ Z5)) \/ Z6) \/ Z7) \/ Z8) \/ Z9 by XBOOLE_1:4;
then Y in (((Z1 \/ ((((Z2 \/ Z3) \/ Z4) \/ Z5) \/ Z6)) \/ Z7) \/ Z8) \/ Z9 by XBOOLE_1:4;
then Y in ((Z1 \/ (((((Z2 \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7)) \/ Z8) \/ Z9 by XBOOLE_1:4;
then Y in (Z1 \/ ((((((Z2 \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8)) \/ Z9 by XBOOLE_1:4;
then Y in Z1 \/ (((((((Z2 \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) by XBOOLE_1:4;
then Y in ((((((Z2 \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9 by A13, XBOOLE_0:def 3;
then Y in (((((Z2 \/ (Z3 \/ Z4)) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9 by XBOOLE_1:4;
then Y in ((((Z2 \/ ((Z3 \/ Z4) \/ Z5)) \/ Z6) \/ Z7) \/ Z8) \/ Z9 by XBOOLE_1:4;
then Y in (((Z2 \/ (((Z3 \/ Z4) \/ Z5) \/ Z6)) \/ Z7) \/ Z8) \/ Z9 by XBOOLE_1:4;
then Y in ((Z2 \/ ((((Z3 \/ Z4) \/ Z5) \/ Z6) \/ Z7)) \/ Z8) \/ Z9 by XBOOLE_1:4;
then Y in (Z2 \/ (((((Z3 \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8)) \/ Z9 by XBOOLE_1:4;
then Y in Z2 \/ ((((((Z3 \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) by XBOOLE_1:4;
then Y in (((((Z3 \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9 by A18, XBOOLE_0:def 3;
then Y in ((((Z3 \/ (Z4 \/ Z5)) \/ Z6) \/ Z7) \/ Z8) \/ Z9 by XBOOLE_1:4;
then Y in (((Z3 \/ ((Z4 \/ Z5) \/ Z6)) \/ Z7) \/ Z8) \/ Z9 by XBOOLE_1:4;
then Y in ((Z3 \/ (((Z4 \/ Z5) \/ Z6) \/ Z7)) \/ Z8) \/ Z9 by XBOOLE_1:4;
then Y in (Z3 \/ ((((Z4 \/ Z5) \/ Z6) \/ Z7) \/ Z8)) \/ Z9 by XBOOLE_1:4;
then A28: Y in Z3 \/ (((((Z4 \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) by XBOOLE_1:4;
A29: now
assume A30: Y in Z4 ; :: thesis: contradiction
then consider Y1, Y2, Y3, Y4, Y5 being set such that
A31: ( Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 ) and
A32: Y5 in Y and
A33: Y1 meets X by A5;
Y in union (union (union (union X))) by A5, A30;
then Y5 in union (union (union (union (union X)))) by A32, TARSKI:def 4;
then Y5 in Z5 by A4, A31, A33;
then Y5 in ((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5 by XBOOLE_0:def 3;
then Y5 in (((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6 by XBOOLE_0:def 3;
then Y5 in ((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7 by XBOOLE_0:def 3;
then Y5 in (((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8 by XBOOLE_0:def 3;
then Y5 in ((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9 by XBOOLE_0:def 3;
hence contradiction by A11, A32, XBOOLE_0:3; :: thesis: verum
end;
A34: now
assume A35: Y in Z5 ; :: thesis: contradiction
then consider Y1, Y2, Y3, Y4 being set such that
A36: ( Y1 in Y2 & Y2 in Y3 & Y3 in Y4 ) and
A37: Y4 in Y and
A38: Y1 meets X by A4;
Y in union (union (union (union (union X)))) by A4, A35;
then Y4 in union (union (union (union (union (union X))))) by A37, TARSKI:def 4;
then Y4 in Z6 by A9, A36, A38;
then Y4 in (((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6 by XBOOLE_0:def 3;
then Y4 in ((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7 by XBOOLE_0:def 3;
then Y4 in (((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8 by XBOOLE_0:def 3;
then Y4 in ((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9 by XBOOLE_0:def 3;
hence contradiction by A11, A37, XBOOLE_0:3; :: thesis: verum
end;
A39: now
assume A40: Y in Z6 ; :: thesis: contradiction
then consider Y1, Y2, Y3 being set such that
A41: ( Y1 in Y2 & Y2 in Y3 ) and
A42: Y3 in Y and
A43: Y1 meets X by A9;
Y in union (union (union (union (union (union X))))) by A9, A40;
then Y3 in union (union (union (union (union (union (union X)))))) by A42, TARSKI:def 4;
then Y3 in Z7 by A8, A41, A43;
then Y3 in ((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7 by XBOOLE_0:def 3;
then Y3 in (((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8 by XBOOLE_0:def 3;
then Y3 in ((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9 by XBOOLE_0:def 3;
hence contradiction by A11, A42, XBOOLE_0:3; :: thesis: verum
end;
A44: now
assume A45: Y in Z7 ; :: thesis: contradiction
then consider Y1, Y2 being set such that
A46: Y1 in Y2 and
A47: Y2 in Y and
A48: Y1 meets X by A8;
Y in union (union (union (union (union (union (union X)))))) by A8, A45;
then Y2 in union (union (union (union (union (union (union (union X))))))) by A47, TARSKI:def 4;
then Y2 in Z8 by A3, A46, A48;
then Y2 in (((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8 by XBOOLE_0:def 3;
then Y2 in ((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9 by XBOOLE_0:def 3;
hence contradiction by A11, A47, XBOOLE_0:3; :: thesis: verum
end;
A49: now
assume A50: Y in Z8 ; :: thesis: contradiction
then consider Y1 being set such that
A51: Y1 in Y and
A52: Y1 meets X by A3;
Y in union (union (union (union (union (union (union (union X))))))) by A3, A50;
then Y1 in union (union (union (union (union (union (union (union (union X)))))))) by A51, TARSKI:def 4;
then Y1 in Z9 by A6, A52;
then Y1 in ((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9 by XBOOLE_0:def 3;
hence contradiction by A11, A51, XBOOLE_0:3; :: thesis: verum
end;
now
assume A53: Y in Z3 ; :: thesis: contradiction
then consider Y1, Y2, Y3, Y4, Y5, Y6 being set such that
A54: ( Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 ) and
A55: Y6 in Y and
A56: Y1 meets X by A7;
Y in union (union (union X)) by A7, A53;
then Y6 in union (union (union (union X))) by A55, TARSKI:def 4;
then Y6 in Z4 by A5, A54, A56;
then Y6 in (((X \/ Z1) \/ Z2) \/ Z3) \/ Z4 by XBOOLE_0:def 3;
then Y6 in ((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5 by XBOOLE_0:def 3;
then Y6 in (((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6 by XBOOLE_0:def 3;
then Y6 in ((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7 by XBOOLE_0:def 3;
then Y6 in (((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8 by XBOOLE_0:def 3;
then Y6 in ((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9 by XBOOLE_0:def 3;
hence contradiction by A11, A55, XBOOLE_0:3; :: thesis: verum
end;
then Y in ((((Z4 \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9 by A28, XBOOLE_0:def 3;
then Y in (((Z4 \/ (Z5 \/ Z6)) \/ Z7) \/ Z8) \/ Z9 by XBOOLE_1:4;
then Y in ((Z4 \/ ((Z5 \/ Z6) \/ Z7)) \/ Z8) \/ Z9 by XBOOLE_1:4;
then Y in (Z4 \/ (((Z5 \/ Z6) \/ Z7) \/ Z8)) \/ Z9 by XBOOLE_1:4;
then Y in Z4 \/ ((((Z5 \/ Z6) \/ Z7) \/ Z8) \/ Z9) by XBOOLE_1:4;
then Y in (((Z5 \/ Z6) \/ Z7) \/ Z8) \/ Z9 by A29, XBOOLE_0:def 3;
then Y in ((Z5 \/ (Z6 \/ Z7)) \/ Z8) \/ Z9 by XBOOLE_1:4;
then Y in (Z5 \/ ((Z6 \/ Z7) \/ Z8)) \/ Z9 by XBOOLE_1:4;
then Y in Z5 \/ (((Z6 \/ Z7) \/ Z8) \/ Z9) by XBOOLE_1:4;
then Y in ((Z6 \/ Z7) \/ Z8) \/ Z9 by A34, XBOOLE_0:def 3;
then Y in (Z6 \/ (Z7 \/ Z8)) \/ Z9 by XBOOLE_1:4;
then Y in Z6 \/ ((Z7 \/ Z8) \/ Z9) by XBOOLE_1:4;
then Y in (Z7 \/ Z8) \/ Z9 by A39, XBOOLE_0:def 3;
then Y in Z7 \/ (Z8 \/ Z9) by XBOOLE_1:4;
then Y in Z8 \/ Z9 by A44, XBOOLE_0:def 3;
then Y in Z9 by A49, XBOOLE_0:def 3;
then Y meets X by A6;
hence contradiction by A12, A11, XBOOLE_1:70; :: thesis: verum