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 ValuatRing v is domRing-like
  proof
    set R = ValuatRing v;
    assume
A1: K is having_valuation;
    let x, y be Element of R;
    assume that
A2: x*y = 0.R and
A3: x <> 0.R;
    reconsider x1 = x, y1 = y as Element of K by A1,Th51;
A4: 0.R = 0.K by A1,Def12;
A5: x1*y1 = x*y by A1,Th55;
    y1 = x1"*(x1*y1) by A3,A4,Lm7
    .= 0.K by A2,A4,A5;
    hence thesis by A1,Def12;
  end;
