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 Th43:
  K is having_valuation implies normal-valuation(v).Pgenerator(v) = 1
  proof
    set f = normal-valuation(v);
    set a = Pgenerator(v);
    set l = least-positive(rng v);
    assume
A1: K is having_valuation;
    then
A2: v.a = (f.a)*l by Def10;
A3: l in REAL by A1,Lm6;
    l in rng v by A1,Lm5;
    then {l} c= rng v by ZFMISC_1:31;
    then
A4: v"{l} is non empty by RELAT_1:139;
    a = the Element of v"{l} by A1,Def9;
    then v.a in {l} by A4,FUNCT_1:def 7;
    then v.a = l by TARSKI:def 1
    .= 1*l by XXREAL_3:81;
    hence f.a = 1 by A2,A3,XXREAL_3:68;
  end;
