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 being Point of X holds f is_continuous_at x & g is continuous
  implies g*f is_continuous_at x
proof
  let x be Point of X;
  assume
A1: f is_continuous_at x;
  assume g is continuous;
  then g is_continuous_at (f.x);
  hence thesis by A1,Th47;
end;
