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
  for I being maximal Ideal of ValuatRing v holds I = vp(v)
  proof
    assume
A1: K is having_valuation;
    let I be maximal Ideal of ValuatRing v;
    assume
A2: not thesis;
    per cases;
    suppose not I c= vp(v);
      hence contradiction by A1,Th79;
    end;
    suppose I c= vp(v);
      then vp(v) is non proper or I = vp(v) by RING_1:def 3;
      then
A3:   vp(v) = the carrier of ValuatRing v or I = vp(v);
      1.ValuatRing v = 1.K by A1,Def12;
      hence contradiction by A3,A1,A2,Th63;
    end;
  end;
