let K be non empty non degenerated right_complementable distributive Abelian add-associative right_zeroed associative Field-like doubleLoopStr ; :: thesis: for a, b being Element of K
for v being Valuation of K st K is having_valuation & b <> 0. K holds
v . (a / b) = (v . a) - (v . b)
let a, b be Element of K; :: thesis: for v being Valuation of K st K is having_valuation & b <> 0. K holds
v . (a / b) = (v . a) - (v . b)
let v be Valuation of K; :: thesis: ( K is having_valuation & b <> 0. K implies v . (a / b) = (v . a) - (v . b) )
assume A1: K is having_valuation ; :: thesis: ( not b <> 0. K or v . (a / b) = (v . a) - (v . b) )
assume A2: b <> 0. K ; :: thesis: v . (a / b) = (v . a) - (v . b)
thus v . (a / b) = (v . a) + (v . (b ")) by A1, Def8
.= (v . a) - (v . b) by A1, A2, Th21 ; :: thesis: verum