reserve X for set;

theorem Th25:
  for Y,S being non empty set, F being Function of Y,S, M being
  Function of S,ExtREAL st M is nonnegative holds M*F is nonnegative
proof
  let Y,S be non empty set;
  let F be Function of Y,S;
  let M be Function of S,ExtREAL;
  assume
A1: M is nonnegative;
  for n being Element of Y holds 0.<= (M*F).n
  proof
    let n be Element of Y;
    dom (M*F) = Y by FUNCT_2:def 1;
    then (M*F).n = M.(F.n) by FUNCT_1:12;
    hence thesis by A1;
  end;
  hence thesis;
end;
