take f = id the carrier of V; :: thesis: ( f is onto & f is isometric )
thus rng f = the carrier of V by RELAT_1:45; :: according to FUNCT_2:def 3 :: thesis: f is isometric
let x, y be Element of V; :: according to VECTMETR:def 3 :: thesis: dist (x,y) = dist ((f . x),(f . y))
thus dist (x,y) = dist ((f . x),y) by FUNCT_1:18
.= dist ((f . x),(f . y)) by FUNCT_1:18 ; :: thesis: verum