reserve f,g,h for Function,
  A for set;
reserve F for Function,
  B,x,y,y1,y2,z for set;
reserve x,z for object;
reserve X for non empty set,
  Y for set,
  F for BinOp of X,
  f,g,h for Function of Y,X,
  x,x1,x2 for Element of X;
reserve Y for non empty set,
  F for BinOp of X,
  f,g,h for Function of Y,X,
  x,x1,x2 for Element of X;
reserve Y for set,
  F for BinOp of X,
  f,g,h for Function of Y,X,
  x,x1,x2 for Element of X;
reserve Y for non empty set,
  F for BinOp of X,
  f,g,h for Function of Y,X,
  x,x1,x2 for Element of X;
reserve Y for set,
  F for BinOp of X,
  f,g,h for Function of Y,X,
  x,x1,x2 for Element of X;
reserve Y for non empty set,
  F for BinOp of X,
  f,g,h for Function of Y,X,
  x,x1,x2 for Element of X;
reserve Y for set,
  F for BinOp of X,
  f,g,h for Function
  of Y,X,
  x,x1,x2 for Element of X;
reserve y for Element of Y;
reserve Y for non empty set,
  F for BinOp of X,
  f for Function of Y,X,
  x for Element of X,
  y for Element of Y;
reserve a,b,c for set;
reserve x,y,z for object;
reserve Y for set,
        f,g for Function of Y,X,
        x for Element of X,
        y for Element of Y;

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;
