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 non empty Subset of K, x being Element of ValuatRing v st
  I is Ideal of ValuatRing v & x in min(I,v) holds
  I = {x}-Ideal
  proof
    assume
A1: K is having_valuation;
    let I be non empty Subset of K;
    let x be Element of ValuatRing v;
    assume
A2: I is Ideal of ValuatRing v;
    assume
A3: x in min(I,v);
A4: min(I,v) c= I by A1,A2,Th64;
    thus I c= {x}-Ideal
    proof
      let a be object;
      assume
A5:   a in I;
      then reconsider y = a as Element of ValuatRing v by A2;
      reconsider x1 = x, y1 = y as Element of K by A1,Th51;
      per cases by A1,Def12;
      suppose
A6:     x <> 0.K;
        set r1 = y1/x1;
        v.x1 <= v.y1 by A1,A2,A3,A5,Th65;
        then 0 <= v.y1 - v.x1 by Lm4;
        then 0 <= v.r1 by A6,A1,Th22;
        then reconsider r0 = r1 as Element of ValuatRing v
        by A1,Th52;
        x1*r1 = y1*(x1"*x1) by GROUP_1:def 3
        .= y1*1_K by A6,VECTSP_2:9
        .= y1; then
A7:     y = x*r0 by A1,Th55;
        {x}-Ideal = the set of all x*r where r is Element of ValuatRing v
        by IDEAL_1:64;
        hence a in {x}-Ideal by A7;
      end;
      suppose
A8:     x = 0.ValuatRing v;
        then
A9:     {x}-Ideal = {0.ValuatRing v} by IDEAL_1:47;
A10:    0.ValuatRing v = 0.K by A1,Def12; then
A11:    v.x1 = +infty by A1,A8,Def8;
        v.x1 <= v.y1 by A1,A5,A2,A3,Th65; then
        y = 0.ValuatRing v by A1,A10,Th18,A11,XXREAL_0:4;
        hence a in {x}-Ideal by A9,TARSKI:def 1;
      end;
    end;
    {x} c= I by A4,A3,ZFMISC_1:31;
    hence {x}-Ideal c= I by A2,IDEAL_1:def 14;
  end;
