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
  normal-valuation(normal-valuation(v)) = normal-valuation(v)
  proof
    assume
A1: K is having_valuation;
    set f = normal-valuation(v);
    set g = normal-valuation(f);
    let a be Element of K;
    set k = least-positive(rng f);
A2: f.a = (g.a)*k by A1,Def10;
    f.Pgenerator(v) = 1 by A1,Th43;
    then k = 1 by Th34;
    hence thesis by A2,XXREAL_3:81;
  end;
