let X be non empty TopSpace; :: thesis: for F, G being VECTOR of (C_Algebra_of_ContinuousFunctions X)
for f, g being Function of the carrier of X,COMPLEX
for a being 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 (C_Algebra_of_ContinuousFunctions X); :: thesis: for f, g being Function of the carrier of X,COMPLEX
for a being 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 the carrier of X,COMPLEX; :: thesis: for a being 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: ( 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_ContinuousFunctions X is ComplexSubAlgebra of CAlgebra the carrier of X by CC0SP1:2;
reconsider f1 = F, g1 = G as VECTOR of (CAlgebra the carrier of X) 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 the carrier of X holds g . x = a * (f . x)
let x be Element of the carrier of X; :: thesis: g . x = a * (f . x)
g1 = a * f1 by A2, A3, CC0SP1:3;
hence g . x = a * (f . x) by A1, CFUNCDOM:4; :: thesis: verum

end;

assume for x being Element of the carrier of X holds g . x = a * (f . x) ; :: thesis: G = a * F
then g1 = a * f1 by A1, CFUNCDOM:4;
hence G = a * F by A2, CC0SP1:3; :: thesis: verum