theorem
  K is having_valuation implies
  ValuatRing(normal-valuation v) = ValuatRing v
  proof
    assume
A1: K is having_valuation;
    set f = normal-valuation v;
    set R = ValuatRing(v);
    set S = ValuatRing(f);
A2: the carrier of S = NonNegElements f by A1,Def12;
A3: NonNegElements(f) = NonNegElements(v) by A1,Th82;
A4: the addF of R = (the addF of K) | [:NonNegElements v,NonNegElements v:]
    by A1,Def12
    .= the addF of S by A1,A3,Def12;
A5: the multF of R = (the multF of K) | [:NonNegElements v,NonNegElements v:]
    by A1,Def12
    .= the multF of S by A1,A3,Def12;
A6: the ZeroF of R = 0.K by A1,Def12
    .= the ZeroF of S by A1,Def12;
    the OneF of R = 1.K by A1,Def12
    .= the OneF of S by A1,Def12;
    hence thesis by A3,A2,A4,A5,A6,A1,Def12;
  end;
