reserve a,b,p,x,x9,x1,x19,x2,y,y9,y1,y19,y2,z,z9,z1,z2 for object,
   X,X9,Y,Y9,Z,Z9 for set;
reserve A,D,D9 for non empty set;
reserve f,g,h for Function;

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;
