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 Th77:
  K is having_valuation implies
  for x being Element of ValuatRing v st not x in vp(v)
  holds v.x = 0
  proof
    assume
A1: K is having_valuation;
    let x be Element of ValuatRing v;
    reconsider y = x as Element of K by A1,Th51;
    assume not x in vp(v);
    then v.y <= 0 by A1,Th61;
    hence thesis by A1,Th52;
  end;
