theorem Th56:
  f is nonnegative & g is nonnegative implies f+g is nonnegative
proof
  assume that
A1: f is nonnegative and
A2: g is nonnegative;
  for x be object st x in dom(f+g) holds 0 <= (f+g).x
  proof
    let x be object such that
A3: x in dom(f+g);
    0 <= f.x by A1,Th51;
    then
A4: g.x <= f.x + g.x by XREAL_1:31;
    0 <= g.x by A2,Th51;
    hence thesis by A3,A4,VALUED_1:def 1;
  end;
  hence thesis by Th52;
end;
