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 Th50:
  for T, S being 1-sorted, f being Function of T,S st rng f = [#]S
  & f is one-to-one holds f" is one-to-one
proof
  let T, S be 1-sorted, f be Function of T,S;
  assume that
A1: rng f = [#]S and
A2: f is one-to-one;
A3: f is onto by A1,FUNCT_2:def 3;
  (f qua Function)" is one-to-one by A2;
  hence thesis by A2,A3,Def4;
end;
