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 Th55:
  for T, S being 1-sorted, f being Function of T,S, P1 being
  Subset of S st rng f = [#]S & f is one-to-one holds f"P1 = (f").:P1
proof
  let T, S be 1-sorted, f be Function of T, S, P1 be Subset of S;
  assume rng f = [#]S;
  then
A1: f is onto by FUNCT_2:def 3;
  assume
A2: f is one-to-one;
  f"P1 = ((f qua Function)").:P1 by A2,FUNCT_1:85;
  hence thesis by A1,A2,Def4;
end;
