let X be set ; :: thesis: ( X <> {} implies ex Y being set st
( Y in X & ( for 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 Y holds
Y1 misses X ) ) )
defpred S1[ 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 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 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 );
consider Z2 being set such that
A2: for Y being set holds
( Y in Z2 iff ( Y in union (union X) & S2[Y] ) ) from XBOOLE_0:sch 1();
 defpred S3[ 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 );
consider Z3 being set such that
A3: for Y being set holds
( Y in Z3 iff ( Y in union (union (union X)) & S3[Y] ) ) from XBOOLE_0:sch 1();
 defpred S4[ 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 );
consider Z4 being set such that
A4: for Y being set holds
( Y in Z4 iff ( Y in union (union (union (union X))) & S4[Y] ) ) from XBOOLE_0:sch 1();
 defpred S5[ set ] means ex Y1, Y2, Y3 being set st
( Y1 in Y2 & Y2 in Y3 & Y3 in $1 & Y1 meets X );
consider Z5 being set such that
A5: for Y being set holds
( Y in Z5 iff ( Y in union (union (union (union (union X)))) & S5[Y] ) ) from XBOOLE_0:sch 1();
 defpred S6[ set ] means ex Y1, Y2 being set st
( Y1 in Y2 & Y2 in $1 & Y1 meets X );
consider Z6 being set such that
A6: for Y being set holds
( Y in Z6 iff ( Y in union (union (union (union (union (union X))))) & S6[Y] ) ) from XBOOLE_0:sch 1();
 defpred S7[ set ] means ex Y1 being set st
( Y1 in $1 & Y1 meets X );
consider Z7 being set such that
A7: for Y being set holds
( Y in Z7 iff ( Y in union (union (union (union (union (union (union X)))))) & S7[Y] ) ) from XBOOLE_0:sch 1();
 defpred S8[ set ] means $1 meets X;
consider Z8 being set such that
A8: 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;
A9: (((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8 = ((((((X \/ (Z1 \/ Z2)) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8 by XBOOLE_1:4
.= (((((X \/ ((Z1 \/ Z2) \/ Z3)) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8 by XBOOLE_1:4
.= ((((X \/ (((Z1 \/ Z2) \/ Z3) \/ Z4)) \/ Z5) \/ Z6) \/ Z7) \/ Z8 by XBOOLE_1:4
.= (((X \/ ((((Z1 \/ Z2) \/ Z3) \/ Z4) \/ Z5)) \/ Z6) \/ Z7) \/ Z8 by XBOOLE_1:4
.= ((X \/ (((((Z1 \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6)) \/ Z7) \/ Z8 by XBOOLE_1:4
.= (X \/ ((((((Z1 \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7)) \/ Z8 by XBOOLE_1:4
.= X \/ (((((((Z1 \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) by XBOOLE_1:4 ;
assume X <> {} ; :: thesis: ex Y being set st
( Y in X & ( for 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 Y holds
Y1 misses X ) )
then (((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8 <> {} ;
then consider Y being set such that
A10: Y in (((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8 and
A11: Y misses (((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8 by MCART_1:1;
assume A12: for Y being set holds
( not Y in X or 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 Y & not Y1 misses X ) ) ; :: thesis: contradiction
now
assume A13: Y in X ; :: thesis: contradiction
then consider Y1, Y2, Y3, Y4, Y5, Y6, Y7, Y8 being set such that
A14: ( Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in Y & not Y1 misses X ) by A12;
( Y8 in union X & Y1 meets X ) by A13, A14, TARSKI:def 4;
then Y8 in Z1 by A1, A14;
then Y8 in X \/ Z1 by XBOOLE_0:def 3;
then Y8 in (X \/ Z1) \/ Z2 by XBOOLE_0:def 3;
then Y8 in ((X \/ Z1) \/ Z2) \/ Z3 by XBOOLE_0:def 3;
then Y8 in (((X \/ Z1) \/ Z2) \/ Z3) \/ Z4 by XBOOLE_0:def 3;
then Y8 in ((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5 by XBOOLE_0:def 3;
then Y meets ((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5 by A14, 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;
hence contradiction by A11, XBOOLE_1:70; :: thesis: verum
end;
then Y in ((((((Z1 \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8 by A9, A10, XBOOLE_0:def 3;
then Y in (((((Z1 \/ (Z2 \/ Z3)) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8 by XBOOLE_1:4;
then Y in ((((Z1 \/ ((Z2 \/ Z3) \/ Z4)) \/ Z5) \/ Z6) \/ Z7) \/ Z8 by XBOOLE_1:4;
then Y in (((Z1 \/ (((Z2 \/ Z3) \/ Z4) \/ Z5)) \/ Z6) \/ Z7) \/ Z8 by XBOOLE_1:4;
then Y in ((Z1 \/ ((((Z2 \/ Z3) \/ Z4) \/ Z5) \/ Z6)) \/ Z7) \/ Z8 by XBOOLE_1:4;
then Y in (Z1 \/ (((((Z2 \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7)) \/ Z8 by XBOOLE_1:4;
then A15: Y in Z1 \/ ((((((Z2 \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) by XBOOLE_1:4;
now
assume A16: Y in Z1 ; :: thesis: contradiction
then consider Y1, Y2, Y3, Y4, Y5, Y6, Y7 being set such that
A17: ( 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 A1;
Y in union X by A1, A16;
then Y7 in union (union X) by A17, TARSKI:def 4;
then Y7 in Z2 by A2, A17;
then Y7 in (X \/ Z1) \/ Z2 by XBOOLE_0:def 3;
then Y meets (X \/ Z1) \/ Z2 by A17, 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;
hence contradiction by A11, XBOOLE_1:70; :: thesis: verum
end;
then Y in (((((Z2 \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8 by A15, XBOOLE_0:def 3;
then Y in ((((Z2 \/ (Z3 \/ Z4)) \/ Z5) \/ Z6) \/ Z7) \/ Z8 by XBOOLE_1:4;
then Y in (((Z2 \/ ((Z3 \/ Z4) \/ Z5)) \/ Z6) \/ Z7) \/ Z8 by XBOOLE_1:4;
then Y in ((Z2 \/ (((Z3 \/ Z4) \/ Z5) \/ Z6)) \/ Z7) \/ Z8 by XBOOLE_1:4;
then Y in (Z2 \/ ((((Z3 \/ Z4) \/ Z5) \/ Z6) \/ Z7)) \/ Z8 by XBOOLE_1:4;
then A18: Y in Z2 \/ (((((Z3 \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) by XBOOLE_1:4;
now
assume A19: Y in Z2 ; :: thesis: contradiction
then consider Y1, Y2, Y3, Y4, Y5, Y6 being set such that
A20: ( Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y & Y1 meets X ) by A2;
Y in union (union X) by A2, A19;
then Y6 in union (union (union X)) by A20, TARSKI:def 4;
then Y6 in Z3 by A3, A20;
then Y6 in ((X \/ Z1) \/ Z2) \/ Z3 by XBOOLE_0:def 3;
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;
hence contradiction by A11, A20, XBOOLE_0:3; :: thesis: verum
end;
then Y in ((((Z3 \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8 by A18, XBOOLE_0:def 3;
then Y in (((Z3 \/ (Z4 \/ Z5)) \/ Z6) \/ Z7) \/ Z8 by XBOOLE_1:4;
then Y in ((Z3 \/ ((Z4 \/ Z5) \/ Z6)) \/ Z7) \/ Z8 by XBOOLE_1:4;
then Y in (Z3 \/ (((Z4 \/ Z5) \/ Z6) \/ Z7)) \/ Z8 by XBOOLE_1:4;
then A21: Y in Z3 \/ ((((Z4 \/ Z5) \/ Z6) \/ Z7) \/ Z8) by XBOOLE_1:4;
now
assume A22: Y in Z3 ; :: thesis: contradiction
then consider Y1, Y2, Y3, Y4, Y5 being set such that
A23: ( Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y & Y1 meets X ) by A3;
Y in union (union (union X)) by A3, A22;
then Y5 in union (union (union (union X))) by A23, TARSKI:def 4;
then Y5 in Z4 by A4, A23;
then Y5 in (((X \/ Z1) \/ Z2) \/ Z3) \/ Z4 by XBOOLE_0:def 3;
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;
hence contradiction by A11, A23, XBOOLE_0:3; :: thesis: verum
end;
then Y in (((Z4 \/ Z5) \/ Z6) \/ Z7) \/ Z8 by A21, XBOOLE_0:def 3;
then Y in ((Z4 \/ (Z5 \/ Z6)) \/ Z7) \/ Z8 by XBOOLE_1:4;
then Y in (Z4 \/ ((Z5 \/ Z6) \/ Z7)) \/ Z8 by XBOOLE_1:4;
then A24: Y in Z4 \/ (((Z5 \/ Z6) \/ Z7) \/ Z8) by XBOOLE_1:4;
now
assume A25: Y in Z4 ; :: thesis: contradiction
then consider Y1, Y2, Y3, Y4 being set such that
A26: ( Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y & Y1 meets X ) by A4;
Y in union (union (union (union X))) by A4, A25;
then Y4 in union (union (union (union (union X)))) by A26, TARSKI:def 4;
then Y4 in Z5 by A5, A26;
then Y4 in ((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5 by XBOOLE_0:def 3;
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;
hence contradiction by A11, A26, XBOOLE_0:3; :: thesis: verum
end;
then Y in ((Z5 \/ Z6) \/ Z7) \/ Z8 by A24, XBOOLE_0:def 3;
then Y in (Z5 \/ (Z6 \/ Z7)) \/ Z8 by XBOOLE_1:4;
then A27: Y in Z5 \/ ((Z6 \/ Z7) \/ Z8) by XBOOLE_1:4;
now
assume A28: Y in Z5 ; :: thesis: contradiction
then consider Y1, Y2, Y3 being set such that
A29: ( Y1 in Y2 & Y2 in Y3 & Y3 in Y & Y1 meets X ) by A5;
Y in union (union (union (union (union X)))) by A5, A28;
then Y3 in union (union (union (union (union (union X))))) by A29, TARSKI:def 4;
then Y3 in Z6 by A6, A29;
then Y3 in (((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6 by XBOOLE_0:def 3;
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;
hence contradiction by A11, A29, XBOOLE_0:3; :: thesis: verum
end;
then Y in (Z6 \/ Z7) \/ Z8 by A27, XBOOLE_0:def 3;
then A30: Y in Z6 \/ (Z7 \/ Z8) by XBOOLE_1:4;
now
assume A31: Y in Z6 ; :: thesis: contradiction
then consider Y1, Y2 being set such that
A32: ( Y1 in Y2 & Y2 in Y & Y1 meets X ) by A6;
Y in union (union (union (union (union (union X))))) by A6, A31;
then Y2 in union (union (union (union (union (union (union X)))))) by A32, TARSKI:def 4;
then Y2 in Z7 by A7, A32;
then Y2 in ((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7 by XBOOLE_0:def 3;
then Y2 in (((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8 by XBOOLE_0:def 3;
hence contradiction by A11, A32, XBOOLE_0:3; :: thesis: verum
end;
then A33: Y in Z7 \/ Z8 by A30, XBOOLE_0:def 3;
now
assume A34: Y in Z7 ; :: thesis: contradiction
then consider Y1 being set such that
A35: ( Y1 in Y & Y1 meets X ) by A7;
Y in union (union (union (union (union (union (union X)))))) by A7, A34;
then Y1 in union (union (union (union (union (union (union (union X))))))) by A35, TARSKI:def 4;
then Y1 in Z8 by A8, A35;
then Y1 in (((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8 by XBOOLE_0:def 3;
hence contradiction by A11, A35, XBOOLE_0:3; :: thesis: verum
end;
then Y in Z8 by A33, XBOOLE_0:def 3;
then Y meets X by A8;
hence contradiction by A9, A11, XBOOLE_1:70; :: thesis: verum