let X be non empty set ; :: thesis: for a being Complex
for F, G being VECTOR of
for f, g being Function of X,COMPLEX st f = F & g = G holds
( G = a * F iff for x being Element of X holds g . x = a * (f . x) )

let a be Complex; :: thesis: for F, G being VECTOR of
for f, g being Function of X,COMPLEX st f = F & g = G holds
( G = a * F iff for x being Element of X holds g . x = a * (f . x) )

let F, G be VECTOR of ; :: thesis: for f, g being Function of X,COMPLEX st f = F & g = G holds
( G = a * F iff for x being Element of X holds g . x = a * (f . x) )

let f, g be Function of X,COMPLEX; :: thesis: ( f = F & g = G implies ( G = a * F iff for x being Element of X holds g . x = a * (f . x) ) )
assume A1: ( f = F & g = G ) ; :: thesis: ( G = a * F iff for x being Element of X holds g . x = a * (f . x) )
A2: C_Algebra_of_BoundedFunctions X is ComplexSubAlgebra of CAlgebra X by Th2;
reconsider f1 = F, g1 = G as VECTOR of () by TARSKI:def 3;
hereby :: thesis: ( ( for x being Element of X holds g . x = a * (f . x) ) implies G = a * F )
assume A3: G = a * F ; :: thesis: for x being Element of X holds g . x = a * (f . x)
let x be Element of X; :: thesis: g . x = a * (f . x)
g1 = a * f1 by A2, A3, Th3;
hence g . x = a * (f . x) by ; :: thesis: verum
end;
assume for x being Element of X holds g . x = a * (f . x) ; :: thesis: G = a * F
then g1 = a * f1 by ;
hence G = a * F by ; :: thesis: verum