theorem
  K is having_valuation & b is Element of ValuatRing v
  implies v.a <= v.a + v.b
  proof
    assume
A1: K is having_valuation;
    assume b is Element of ValuatRing v;
    then 0 <= v.b by A1,Th52;
    then v.a+0 <= v.a+v.b by XXREAL_3:35;
    hence thesis by XXREAL_3:4;
  end;
