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 Th79:
  K is having_valuation implies
  for I being proper Ideal of ValuatRing v holds I c= vp(v)
  proof
    assume
A1: K is having_valuation;
    let I be proper Ideal of ValuatRing v;
A2: I <> the carrier of ValuatRing v by SUBSET_1:def 6;
    assume not I c= vp(v);
    then consider x being object such that
A3: x in I and
A4: not x in vp(v);
A5: x is Element of K by A1,A3,Th51;
    v.x = 0 by A1,A4,A3,Th77;
    hence thesis by A1,A2,A3,A5,Th74;
  end;
