let X be non empty TopSpace; for F, G, H being VECTOR of (C_Algebra_of_ContinuousFunctions X)
for f, g, h being Function of the carrier of X,COMPLEX st f = F & g = G & h = H holds
( H = F * G iff for x being Element of the carrier of X holds h . x = (f . x) * (g . x) )
let F, G, H be VECTOR of (C_Algebra_of_ContinuousFunctions X); for f, g, h being Function of the carrier of X,COMPLEX st f = F & g = G & h = H holds
( H = F * G iff for x being Element of the carrier of X holds h . x = (f . x) * (g . x) )
let f, g, h be Function of the carrier of X,COMPLEX; ( f = F & g = G & h = H implies ( H = F * G iff for x being Element of the carrier of X holds h . x = (f . x) * (g . x) ) )
assume A1:
( f = F & g = G & h = H )
; ( H = F * G iff for x being Element of the carrier of X holds h . x = (f . x) * (g . x) )
A2:
C_Algebra_of_ContinuousFunctions X is ComplexSubAlgebra of CAlgebra the carrier of X
by CC0SP1:2;
reconsider f1 = F, g1 = G, h1 = H as VECTOR of (CAlgebra the carrier of X) by TARSKI:def 3;
hereby ( ( for x being Element of the carrier of X holds h . x = (f . x) * (g . x) ) implies H = F * G )
end;
assume
for x being Element of X holds h . x = (f . x) * (g . x)
; H = F * G
then
h1 = f1 * g1
by A1, CFUNCDOM:2;
hence
H = F * G
by A2, CC0SP1:3; verum