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 & a <> 0.K &
  a is Element of ValuatRing v & a" is Element of ValuatRing v implies
  v.a = 0
  proof
    assume
A1: K is having_valuation;
    assume that
A2: a <> 0.K and
A3: a is Element of ValuatRing v;
    assume a" is Element of ValuatRing v;
    then 0 <= v.a" by A1,Th52;
    then --v.a <= -0 by A1,A2,Th21;
    hence thesis by A3,A1,Th52;
  end;
