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 Th30:
  K is having_valuation & 0 <= v.(1.K+a) implies 0 <= v.a
  proof
    assume that
A1: K is having_valuation & 0 <= v.(1.K+a) and
A2: v.a < 0;
    0 = v.1.K by A1,Th17;
    hence contradiction by A1,A2,Th28;
  end;
