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 Th34:
  v.a = 1 implies least-positive(rng v) = 1
  proof
    assume
A1: v.a = 1;
    dom v = the carrier of K by FUNCT_2:def 1;
    then
A2: v.a in rng v by FUNCT_1:def 3;
    now
      let i be positive ExtInt;
      assume i in rng v;
      per cases by XXREAL_3:1;
      suppose i is positive Real;
        then reconsider i1 = i as positive Real;
        ex p, q being Real st p = 1. & q = i & p <= q
        proof
          reconsider jj=1, i1 as Real;
          take jj, i1;
          0+1 <= i1 by INT_1:7;
          hence thesis;
        end;
        hence 1. <= i;
      end;
      suppose i = +infty;
        hence 1. <= i by XXREAL_0:3;
      end;
    end;
    hence thesis by A1,A2,Def2;
  end;
