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 Th51:
  K is having_valuation implies
  for x being Element of ValuatRing v holds x is Element of K
  proof
    assume
A1: K is having_valuation;
    let x be Element of ValuatRing v;
    the carrier of ValuatRing v = NonNegElements v by A1,Def12; then
    x in NonNegElements v & NonNegElements v c= the carrier of K by Th48;
    hence thesis;
  end;
