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

0. K in vp v

let v be Valuation of K; :: thesis: ( K is having_valuation implies 0. K in vp v )

assume A1: K is having_valuation ; :: thesis: 0. K in vp v

then vp v = { x where x is Element of K : 0 < v . x } by Def13;

hence 0. K in vp v by A1, Lm9; :: thesis: verum

0. K in vp v

let v be Valuation of K; :: thesis: ( K is having_valuation implies 0. K in vp v )

assume A1: K is having_valuation ; :: thesis: 0. K in vp v

then vp v = { x where x is Element of K : 0 < v . x } by Def13;

hence 0. K in vp v by A1, Lm9; :: thesis: verum