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 Th53:
  for T being 1-sorted, S, V being non empty 1-sorted, f being
  Function of T,S, g being Function of S,V st rng f = [#]S & f is one-to-one &
  dom g = [#]S & rng g = [#]V & g is one-to-one holds (g*f)" = f"*g"
proof
  let T be 1-sorted, S, V be non empty 1-sorted;
  let f be Function of T,S;
  let g be Function of S,V;
  assume that
A1: rng f = [#]S and
A2: f is one-to-one;
  assume that
A3: dom g = [#]S and
A4: rng g = [#] V and
A5: g is one-to-one;
A6: f is onto & g is onto by A1,A4,FUNCT_2:def 3;
    rng(g*f) = [#] V by A1,A3,A4,RELAT_1:28;
    then g*f is onto by FUNCT_2:def 3;
  then
A7: (g*f)" = ((g*f) qua Function)" by A2,A5,Def4;
A8: f" = (f qua Function)" by A2,A6,Def4;
  g" = (g qua Function)" by A5,A6,Def4;
  hence thesis by A2,A5,A8,A7,FUNCT_1:44;
end;
