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 Th20:
  K is having_valuation implies v.-a = v.a
  proof
    assume
A1: K is having_valuation;
    (-1.K) * a = -a by VECTSP_2:29;
    hence v.-a = v.(-1.K) + v.a by A1,Def8
    .= 0 + v.a by A1,Th19
    .= v.a by XXREAL_3:4;
  end;
