theorem Th47:
  for x being Point of X, y being Point of Y st y = f.x holds f
  is_continuous_at x & g is_continuous_at y implies g*f is_continuous_at x
proof
  let x be Point of X, y be Point of Y such that
A1: y = f.x;
  assume that
A2: f is_continuous_at x and
A3: g is_continuous_at y;
  for G being a_neighborhood of (g*f).x holds (g*f)"G is a_neighborhood of x
  proof
    let G be a_neighborhood of (g*f).x;
    (g*f).x = g.y by A1,FUNCT_2:15;
    then g"G is a_neighborhood of f.x by A1,A3,Th42;
    then f"(g"G) is a_neighborhood of x by A2,Th42;
    hence thesis by RELAT_1:146;
  end;
  hence thesis by Th42;
end;
