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
  K is having_valuation implies
  for x being Element of K, y being Element of ValuatRing v
  st x = y holds -x = -y
  proof
    set R = ValuatRing v;
    set c = NonNegElements v;
    assume
A1: K is having_valuation;
    let x be Element of K,
        y be Element of R such that
A2: x = y;
A3: 0 <= v.y by A1,A2,Th52;
    v.-x = v.x by A1,Th20;
    then reconsider x1 = -x as Element of R by A1,A2,A3,Th52;
    x+-x = 0.K by RLVECT_1:def 10;
    then y+x1 = 0.K by A2,A1,Th54
    .= 0.ValuatRing v by A1,Def12;
    hence thesis by RLVECT_1:def 10;
  end;
