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 Th21:
  K is having_valuation & a <> 0.K implies v.(a") = -v.a
  proof
    assume that
A1: K is having_valuation and
A2: a <> 0.K;
    a * a" = 1.K by A2,VECTSP_2:def 2;
    then
A3: v.a + v.(a") = v.(1.K) by A1,Def8
    .= 0 by A1,Th17;
    now
      assume a" = 0.K;
      then 1.K = a*0.K by A2,VECTSP_2:def 2
      .= 0.K;
      hence contradiction;
    end;
    then v.a in INT & v.(a") in INT by A1,A2,Def8;
    then reconsider w1 = v.a, w2 = v.(a") as Element of REAL by XREAL_0:def 1;
    w1 + w2 = 0 by A3,XXREAL_3:def 2;
    then -w1 = w2;
    hence thesis by XXREAL_3:def 3;
  end;
