 reserve x,y,z,t for object,X,Y,Z,W for set;
 reserve R,S,T for Relation;

theorem
  (X misses Y & R c= [:X,Y:] \/ [:Y,X:] & [x,y] in R & x in X implies
    not x in Y & not y in X & y in Y) &
  (X misses Y & R c= [:X,Y:] \/ [:Y,X:] & [x,y] in R & y in Y implies
    not y in X & not x in Y & x in X) &
  (X misses Y & R c= [:X,Y:] \/ [:Y,X:] & [x,y] in R & x in Y implies
    not x in X & not y in Y & y in X) &
  (X misses Y & R c= [:X,Y:] \/ [:Y,X:] & [x,y] in R & y in X implies
    not x in X & not y in Y & x in Y)
proof
  thus X misses Y & R c= [:X,Y:] \/ [:Y,X:] & [x,y] in R & x in X implies not
  x in Y & not y in X & y in Y
  proof
    assume that
A1: X misses Y and
A2: R c= [:X,Y:] \/ [:Y,X:] & [x,y] in R and
A3: x in X;
A4: not [x,y] in [:Y,X:]
    proof
      assume
A5:   [x,y] in [:Y,X:];
      not x in Y by A1,A3,XBOOLE_0:3;
      hence thesis by A5,ZFMISC_1:87;
    end;
A6: [x,y] in [:X,Y:] implies thesis
    proof
      assume [x,y] in [:X,Y:];
      then x in X & y in Y by ZFMISC_1:87;
      hence thesis by A1,XBOOLE_0:3;
    end;
    [:X,Y:] misses [:Y,X:] by A1,ZFMISC_1:104;
    hence thesis by A2,A6,A4,XBOOLE_0:5;
  end;
  thus X misses Y & R c= [:X,Y:] \/ [:Y,X:] & [x,y] in R & y in Y implies
  not y in X & not x in Y & x in X
  proof
    assume that
A7: X misses Y and
A8: R c= [:X,Y:] \/ [:Y,X:] & [x,y] in R and
A9: y in Y;
A10: not [x,y] in [:Y,X:]
    proof
      assume
A11:  [x,y] in [:Y,X:];
      not y in X by A7,A9,XBOOLE_0:3;
      hence thesis by A11,ZFMISC_1:87;
    end;
    [x,y] in [:X,Y:] implies thesis
    proof
      assume [x,y] in [:X,Y:]; then
      x in X & y in Y by ZFMISC_1:87;
      hence thesis by A7,XBOOLE_0:3;
    end;
    hence thesis by A8,A10,XBOOLE_0:def 3;
  end;
  thus X misses Y & R c= [:X,Y:] \/ [:Y,X:] & [x,y] in R & x in Y implies not
  x in X & not y in Y & y in X
  proof
    assume that
A12: X misses Y and
A13: R c= [:X,Y:] \/ [:Y,X:] & [x,y] in R and
A14: x in Y;
A15: not [x,y] in [:X,Y:]
    proof
      assume
A16:  [x,y] in [:X,Y:];
      not x in X by A12,A14,XBOOLE_0:3;
      hence thesis by A16,ZFMISC_1:87;
    end;
    [x,y] in [:Y,X:] & not [x,y] in [:X,Y:] implies thesis
    proof
      assume [x,y] in [:Y,X:] & not [x,y] in [:X,Y:]; then
      x in Y & y in X & not x in X or x in Y & y in X & not y in Y
        by ZFMISC_1:87;
      hence thesis by A12,XBOOLE_0:3;
    end;
    hence thesis by A13,A15,XBOOLE_0:def 3;
  end;
    assume that
A17: X misses Y and
A18: R c= [:X,Y:] \/ [:Y,X:] & [x,y] in R and
A19: y in X;
A20: not [x,y] in [:X,Y:]
    proof
      assume
A21:  [x,y] in [:X,Y:];
      not y in Y by A17,A19,XBOOLE_0:3;
      hence thesis by A21,ZFMISC_1:87;
    end;
    [x,y] in [:Y,X:] implies thesis
    proof
      assume [x,y] in [:Y,X:]; then
      x in Y & y in X by ZFMISC_1:87;
      hence thesis by A17,XBOOLE_0:3;
    end;
    hence thesis by A18,A20,XBOOLE_0:def 3;
end;
