let K be non empty non degenerated right_complementable distributive Abelian add-associative right_zeroed associative Field-like doubleLoopStr ; :: thesis: for a being Element of K
for v being Valuation of K st K is having_valuation holds
( 0 <= v . a iff a is Element of (ValuatRing v) )
let a be Element of K; :: thesis: for v being Valuation of K st K is having_valuation holds
( 0 <= v . a iff a is Element of (ValuatRing v) )
let v be Valuation of K; :: thesis: ( K is having_valuation implies ( 0 <= v . a iff a is Element of (ValuatRing v) ) )
assume K is having_valuation ; :: thesis: ( 0 <= v . a iff a is Element of (ValuatRing v) )
then the carrier of (ValuatRing v) = NonNegElements v by Def12;
hence ( 0 <= v . a iff a is Element of (ValuatRing v) ) by Th47; :: thesis: verum