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 Th35:
  K is having_valuation implies least-positive(rng v) is integer
  proof
    set l = least-positive(rng v);
    assume
A1: K is having_valuation;
    then consider a such that
A2: v.a <> 0 and
A3: v.a <> +infty by Def8;
A4: dom v = the carrier of K by FUNCT_2:def 1;
    then
A5: v.a in rng v by FUNCT_1:def 3;
    assume not thesis;
    then
A6: l = +infty by Def1;
A7: a <> 0.K by A1,A3,Def8;
    then v.a in INT by A1,Def8;
    then reconsider va = v.a as Real;
    per cases;
    suppose va is positive;
      then l <= v.a by A5,Def2;
      hence contradiction by A3,A6,XXREAL_0:4;
    end;
    suppose not va is positive;
      then reconsider va as non positive Real;
      reconsider va as negative Real by A2;
      set b = a";
      b <> 0.K by A7,VECTSP_2:13;
      then
A8:   v.b in INT by A1,Def8;
A9:  v.b in rng v by A4,FUNCT_1:def 3;
      v.b = -v.a by A1,A7,Th21
      .= -va by XXREAL_3:def 3;
      then l <= v.b by A9,Def2;
      hence contradiction by A8,A6,XXREAL_0:4;
    end;
  end;
