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
not 1. K in vp v
let v be Valuation of K; :: thesis: ( K is having_valuation implies not 1. K in vp v )
assume A1: K is having_valuation ; :: thesis: not 1. K in vp v
then A2: vp v = { x where x is Element of K : 0 < v . x } by Def13;
assume 1. K in vp v ; :: thesis: contradiction
then ex x being Element of K st
( x = 1. K & 0 < v . x ) by A2;
hence contradiction by A1, Th17; :: thesis: verum