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 implies 0.K in vp v
  proof
    assume
A1: K is having_valuation;
    then vp v = {x where x is Element of K: 0 < v.x} by Def13;
    hence thesis by A1,Lm9;
  end;
