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
  for T, S, V being non empty TopStruct, f being Function of T,S, g
  being Function of S,V st f is being_homeomorphism & g is being_homeomorphism
  holds g*f is being_homeomorphism
proof
  let T, S, V be non empty TopStruct;
  let f be Function of T,S;
  let g be Function of S,V;
  assume that
A1: f is being_homeomorphism and
A2: g is being_homeomorphism;
A3: rng f = [#](S) & dom g = [#]S by A1,A2;
A4: rng g = [#](V) by A2;
  dom f = [#]T by A1;
  hence dom(g*f) = [#]T & rng(g*f) = [#] V by A3,A4,RELAT_1:27,28;
A5: f is one-to-one & g is one-to-one by A1,A2;
  hence g*f is one-to-one;
  f is continuous & g is continuous by A1,A2;
  hence g*f is continuous by Th46;
  f" is continuous & g" is continuous by A1,A2;
  then f"*g" is continuous by Th46;
  hence thesis by A3,A4,A5,Th53;
end;
