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 Th46:
  for T,V being TopStruct,S being non empty TopStruct, f being
  Function of T,S, g being Function of S,V holds f is continuous & g is
  continuous implies g*f is continuous
proof
  let T,V be TopStruct,S be non empty TopStruct;
  let f be Function of T,S, g be Function of S,V;
  assume that
A1: f is continuous and
A2: g is continuous;
  let P be Subset of V;
  assume P is closed;
  then (g*f)"P = f"(g"P) & g"P is closed by A2,RELAT_1:146;
  hence thesis by A1;
end;
