theorem
  for X,Y being set, x,y being object
  holds X --> x tolerates Y --> y iff x = y or X misses Y
proof
  let X,Y be set;
  let x,y be object;
  set f = X --> x, g = Y --> y;
  thus f tolerates g implies x = y or X misses Y
  proof
    assume that
A3: for z being object st z in dom f /\ dom g holds f.z = g.z and
A4: x <> y;
    assume X meets Y;
    then consider z be object such that
A5: z in X and
A6: z in Y by XBOOLE_0:3;
A7: f.z = x by A5,Th7;
A8: g.z = y by A6,Th7;
    z in X /\ Y by A5,A6,XBOOLE_0:def 4;
    hence thesis by A3,A4,A7,A8;
  end;
  assume
A9: x = y or X misses Y;
  let z be object;
  assume
A10: z in dom f /\ dom g;
  then
A11: z in Y;
A12: z in X by A10,XBOOLE_0:def 4;
  then f.z = x by Th7;
  hence thesis by A9,A12,A11,Th7,XBOOLE_0:3;
end;
