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 Th72:
  K is having_valuation & v.a = 0 implies
  a is Element of ValuatRing v & a" is Element of ValuatRing v
  proof
    assume
A1: K is having_valuation;
    assume
A2: v.a = 0;
    thus a is Element of ValuatRing v by A1,A2,Th52;
    a <> 0.K by A1,A2,Def8;
    then v.a" = -v.a by A1,Th21;
    hence thesis by A1,A2,Th52;
  end;
