reserve x, y for set,
  T for TopStruct,
  GX for TopSpace,
  P, Q, M, N for Subset of T,
  F, G for Subset-Family of T,
  W, Z for Subset-Family of GX,
  A for SubSpace of T;
reserve S for non empty TopStruct,
  f for Function of T, S,
  H for Subset-Family of S;

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;
