theorem
 for a,b being object holds
  X <> {} & X --> a c= Y --> b implies a = b
proof let a,b be object;
  assume
A1: X <> {};
  set x = the Element of X;
  assume
A2: X --> a c= Y --> b;
  then X c= Y by Th5;
  then x in Y by A1;
  then
A3: (Y --> b).x = b by FUNCOP_1:7;
  dom(X --> a) = X & (X --> a).x = a by A1,FUNCOP_1:7;
  hence thesis by A1,A2,A3,GRFUNC_1:2;
end;
