 reserve x,y,X1,X2,X3,X4,X5,X6,Y,Y1,Y2,Y3,Y4,Y5,Z,Z1,Z2,Z3,Z4,Z5 for set;
 reserve X for non empty set;

theorem
  ex Y st Y in X &
   for Y1,Y2,Y3,Y4 st Y1 in Y2 & Y2 in  Y3 & Y3 in Y4 & Y4 in Y
    holds Y1 misses X
proof
  defpred P[set] means ex Y1,Y2,Y3 st Y1 in Y2 & Y2 in Y3 & Y3 in $1 & Y1
  meets X;
  consider Z1 such that
A1: for Y holds Y in Z1 iff Y in union X & P[Y] from XFAMILY:sch 1;
  defpred S[set] means $1 meets X;
  defpred R[set] means ex Y1 st Y1 in $1 & Y1 meets X;
  defpred Q[set] means ex Y1,Y2 st Y1 in Y2 & Y2 in $1 & Y1 meets X;
  consider Z2 such that
A2: for Y holds Y in Z2 iff Y in union union X & Q[Y] from XFAMILY:sch
  1;
  consider Z4 such that
A3: for Y holds Y in Z4 iff Y in union union union union X & S[Y] from
  XFAMILY:sch 1;
  consider Z3 such that
A4: for Y holds Y in Z3 iff Y in union union union X & R[Y] from
  XFAMILY:sch 1;
  consider Y such that
A5: Y in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 and
A6: Y misses X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by Th1;
A7: now
    assume
A8: Y in Z3;
    then consider Y1 such that
A9: Y1 in Y and
A10: Y1 meets X by A4;
    Y in union union union X by A4,A8;
    then Y1 in union union union union X by A9,TARSKI:def 4;
    then Y1 in Z4 by A3,A10;
    then Y1 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_0:def 3;
    hence contradiction by A6,A9,XBOOLE_0:3;
  end;
A11: now
    assume
A12: Y in Z1;
    then consider Y1,Y2,Y3 such that
A13: Y1 in Y2 & Y2 in Y3 and
A14: Y3 in Y and
A15: Y1 meets X by A1;
    Y in union X by A1,A12;
    then Y3 in union union X by A14,TARSKI:def 4;
    then Y3 in Z2 by A2,A13,A15;
    then Y3 in X \/ Z1 \/ Z2 by XBOOLE_0:def 3;
    then Y meets X \/ Z1 \/ Z2 by A14,XBOOLE_0:3;
    then Y meets X \/ Z1 \/ Z2 \/ Z3 by XBOOLE_1:70;
    hence contradiction by A6,XBOOLE_1:70;
  end;
A16: X \/ Z1 \/ Z2 \/ Z3 \/ Z4 = X \/ (Z1 \/ Z2) \/ Z3 \/ Z4 by XBOOLE_1:4
    .= X \/ (Z1 \/ Z2 \/ Z3) \/ Z4 by XBOOLE_1:4
    .= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4) by XBOOLE_1:4;
A17: now
    assume
A18: Y in Z2;
    then consider Y1,Y2 such that
A19: Y1 in Y2 and
A20: Y2 in Y and
A21: Y1 meets X by A2;
    Y in union union X by A2,A18;
    then Y2 in union union union X by A20,TARSKI:def 4;
    then Y2 in Z3 by A4,A19,A21;
    then Y2 in X \/ Z1 \/ Z2 \/ Z3 by XBOOLE_0:def 3;
    then Y meets X \/ Z1 \/ Z2 \/ Z3 by A20,XBOOLE_0:3;
    hence contradiction by A6,XBOOLE_1:70;
  end;
  assume
A22: not thesis;
  now
    assume
A23: Y in X;
    then consider Y1,Y2,Y3,Y4 such that
A24: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 and
A25: Y4 in Y and
A26: not Y1 misses X by A22;
    Y4 in union X by A23,A25,TARSKI:def 4;
    then Y4 in Z1 by A1,A24,A26;
    then Y4 in X \/ Z1 by XBOOLE_0:def 3;
    then Y4 in X \/ Z1 \/ Z2 by XBOOLE_0:def 3;
    then Y4 in X \/ Z1 \/ Z2 \/ Z3 by XBOOLE_0:def 3;
    then Y4 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_0:def 3;
    hence contradiction by A6,A25,XBOOLE_0:3;
  end;
  then Y in Z1 \/ Z2 \/ Z3 \/ Z4 by A16,A5,XBOOLE_0:def 3;
  then Y in Z1 \/ (Z2 \/ Z3) \/ Z4 by XBOOLE_1:4;
  then Y in Z1 \/ (Z2 \/ Z3 \/ Z4) by XBOOLE_1:4;
  then Y in Z2 \/ Z3 \/ Z4 by A11,XBOOLE_0:def 3;
  then Y in Z2 \/ (Z3 \/ Z4) by XBOOLE_1:4;
  then Y in Z3 \/ Z4 by A17,XBOOLE_0:def 3;
  then Y in Z4 by A7,XBOOLE_0:def 3;
  then Y meets X by A3;
  hence contradiction by A16,A6,XBOOLE_1:70;
end;
