reserve A, B for non empty set,
  A1, A2, A3 for non empty Subset of A;
reserve X for TopSpace;
reserve X for non empty TopSpace;
reserve X1, X2 for non empty SubSpace of X;
reserve X0, X1, X2, Y1, Y2 for non empty SubSpace of X;
reserve X, Y for non empty TopSpace;
reserve f for Function of X,Y;
reserve X,Y,Z for non empty TopSpace;
reserve f for Function of X,Y,
  g for Function of Y,Z;

theorem
  for X,Y,Z being non empty TopSpace st the carrier of X = the carrier
  of Y & the topology of Y c= the topology of X for f being continuous Function
  of Y,Z holds f is continuous Function of X,Z
proof
  let X,Y,Z be non empty TopSpace;
  assume that
A1: the carrier of X = the carrier of Y and
A2: the topology of Y c= the topology of X;
  let f be continuous Function of Y,Z;
  reconsider g = f as Function of X,Z by A1;
  for x being Point of X holds g is_continuous_at x
  by A1,Th44,A2,Th46;
  hence thesis by Th44;
end;
