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;
reserve X, Y for non empty TopSpace,
  X0 for non empty SubSpace of X;
reserve f for Function of X,Y;
reserve f for Function of X,Y,
  X0 for non empty SubSpace of X;

theorem Th62:
  for X,Y,Z being non empty TopSpace, X0 being non empty SubSpace
of X, f being Function of X,Y, g being Function of Y,Z holds (g*f)|X0 = g*(f|X0
  )
proof
  let X,Y,Z be non empty TopSpace, X0 be non empty SubSpace of X, f be
  Function of X,Y, g be Function of Y,Z;
  set h = g*f;
  h|X0 = h|the carrier of X0;
  then reconsider G = h|the carrier of X0 as Function of X0,Z;
  f|X0 = f|the carrier of X0;
  then reconsider F0 = f|the carrier of X0 as Function of X0,Y;
  set F = g*F0;
  for x being Point of X0 holds G.x = F.x
  proof
    let x be Point of X0;
    the carrier of X0 c= the carrier of X by BORSUK_1:1;
    then reconsider y = x as Element of X by TARSKI:def 3;
    thus G.x = (g*f).y by FUNCT_1:49
      .= g.(f.y) by FUNCT_2:15
      .= g.(F0.x) by FUNCT_1:49
      .= F.x by FUNCT_2:15;
  end;
  hence thesis by FUNCT_2:63;
end;
