let K be non empty non degenerated right_complementable distributive Abelian add-associative right_zeroed associative Field-like doubleLoopStr ; :: thesis: for v being Valuation of K st K is having_valuation holds
vp v is proper
let v be Valuation of K; :: thesis: ( K is having_valuation implies vp v is proper )
assume A1: K is having_valuation ; :: thesis: vp v is proper
then 1. K is Element of (ValuatRing v) by Def12;
hence vp v <> the carrier of (ValuatRing v) by A1, Th63; :: according to SUBSET_1:def 6 :: thesis: verum