let x be Vector of V; :: according to HERMITAN:def 1 :: thesis: for a being Scalar of holds (0Functional V) . (a * x) = (a *') * ((0Functional V) . x)

let r be Scalar of ; :: thesis: (0Functional V) . (r * x) = (r *') * ((0Functional V) . x)

(0Functional V) . x = 0. F_Complex by HAHNBAN1:14;

hence (0Functional V) . (r * x) = (r *') * ((0Functional V) . x) by HAHNBAN1:14; :: thesis: verum

let r be Scalar of ; :: thesis: (0Functional V) . (r * x) = (r *') * ((0Functional V) . x)

(0Functional V) . x = 0. F_Complex by HAHNBAN1:14;

hence (0Functional V) . (r * x) = (r *') * ((0Functional V) . x) by HAHNBAN1:14; :: thesis: verum