reserve x, y, z, s for ExtReal;
reserve i, j for Integer;
reserve n, m for Nat;
reserve x, y, v, u for ExtInt;
reserve
  D for non empty doubleLoopStr,
  A for Subset of D;
reserve K for Field-like non degenerated
  associative add-associative right_zeroed right_complementable
  distributive Abelian non empty doubleLoopStr,
  a, b, c for Element of K;
reserve v for Valuation of K;

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;
