theorem MinMax:
  for a,b be Real holds
    a + b = min(a,b) + max(a,b)
  proof
    let a,b be Real;
    per cases;
    suppose a >= b; then
      a = max(a,b) & b = min(a,b) by XXREAL_0:def 9,def 10;
      hence thesis;
    end;
    suppose a < b; then
      a = min(a,b) & b = max(a,b) by XXREAL_0:def 9,def 10;
      hence thesis;
    end;
  end;
