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 Th51:
  for X,Y,Z being non empty TopSpace st the carrier of Y = the
  carrier of Z & the topology of Z c= the topology of Y for f being continuous
  Function of X,Y holds f is continuous Function of X,Z
proof
  let X,Y,Z be non empty TopSpace;
  assume that
A1: the carrier of Y = the carrier of Z and
A2: the topology of Z c= the topology of Y;
  let f be continuous Function of X,Y;
  reconsider g = f as Function of X,Z by A1;
  for x being Point of X holds g is_continuous_at x
  by Th44,A1,A2,Th45;
  hence thesis by Th44;
end;
