let F be Field; :: thesis: for x being Element of F
for V being VectSp of F
for v being Element of V st x <> 0. F holds
(x ") * (x * v) = v
let x be Element of F; :: thesis: for V being VectSp of F
for v being Element of V st x <> 0. F holds
(x ") * (x * v) = v
let V be VectSp of F; :: thesis: for v being Element of V st x <> 0. F holds
(x ") * (x * v) = v
let v be Element of V; :: thesis: ( x <> 0. F implies (x ") * (x * v) = v )
assume A1: x <> 0. F ; :: thesis: (x ") * (x * v) = v
(x ") * (x * v) = ((x ") * x) * v by Def15
.= (1. F) * v by A1, Def10
.= v by Def16 ;
hence (x ") * (x * v) = v ; :: thesis: verum