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 Th61:
  K is having_valuation implies
  (a in vp v iff 0 < v.a)
  proof
    assume K is having_valuation;
    then
A1: vp v = {x where x is Element of K: 0 < v.x} by Def13;
    hereby
      assume a in vp v;
      then ex b being Element of K st b = a & 0 < v.b by A1;
      hence 0 < v.a;
    end;
    thus thesis by A1;
  end;
