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 y being Point of Y holds f is continuous & g is_continuous_at y
  implies for x being Point of X st x in f"{y} holds g*f is_continuous_at x
proof
  let y be Point of Y;
  assume
A1: f is continuous;
  assume
A2: g is_continuous_at y;
  let x be Point of X;
  assume x in f"{y};
  then {x} is Subset of f"{y} by SUBSET_1:41;
  then dom f = [#]X & Im(f,x) c= {y} by FUNCT_2:95,def 1;
  then
A3: {f.x} c= {y} by FUNCT_1:59;
  f.x in {f.x} by TARSKI:def 1;
  then
A4: f.x = y by A3,TARSKI:def 1;
  f is_continuous_at x by A1;
  hence thesis by A2,A4,Th47;
end;
