theorem Th54:
  for T, S being 1-sorted, f being Function of T, S, P being
  Subset of T st rng f = [#]S & f is one-to-one holds f.:P = (f")"P
proof
  let T, S be 1-sorted, f be Function of T, S, P be Subset of T;
  assume that
A1: rng f = [#]S and
A2: f is one-to-one;
A3: f is onto by A1,FUNCT_2:def 3;
  f.:P = ((f qua Function)")"P by A2,FUNCT_1:84;
  hence thesis by A2,A3,Def4;
end;
