reserve X,Y,Z,X1,X2,Y1,Y2 for set, x,y,z,t,x1,x2 for object,
  f,g,h,f1,f2,g1,g2 for Function;

theorem
  rng f1 c= Funcs(X,Y) & rng f2 c= Funcs(X,Y) & X <> {} & uncurry' f1 =
  uncurry' f2 implies f1 = f2
proof
  assume that
A1: rng f1 c= Funcs(X,Y) and
A2: rng f2 c= Funcs(X,Y) and
A3: X <> {} & uncurry' f1 = uncurry' f2;
  dom uncurry f1 = [:dom f1,X:] by A1,Th19;
  then
A4: uncurry f1 = ~~(uncurry f1) by FUNCT_4:52;
  dom uncurry f2 = [:dom f2,X:] by A2,Th19;
  hence thesis by A1,A2,A3,A4,Th39,FUNCT_4:52;
end;
