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 & v . a = 1 holds
normal-valuation v = v
let a be Element of K; :: thesis: for v being Valuation of K st K is having_valuation & v . a = 1 holds
normal-valuation v = v
let v be Valuation of K; :: thesis: ( K is having_valuation & v . a = 1 implies normal-valuation v = v )
set f = normal-valuation v;
assume that
A1: K is having_valuation and
A2: v . a = 1 ; :: thesis: normal-valuation v = v
let a be Element of K; :: according to FUNCT_2:def 8 :: thesis: (normal-valuation v) . a = v . a
thus v . a = ((normal-valuation v) . a) * (least-positive (rng v)) by A1, Def10
.= ((normal-valuation v) . a) * 1 by A2, Th34
.= (normal-valuation v) . a by XXREAL_3:81 ; :: thesis: verum