let X be non empty set ; :: thesis: for f, g, h being Function of X,COMPLEX

for F, G, H being Point of (C_Normed_Algebra_of_BoundedFunctions X) st f = F & g = G & h = H holds

( H = F * G iff for x being Element of X holds h . x = (f . x) * (g . x) )

let f, g, h be Function of X,COMPLEX; :: thesis: for F, G, H being Point of (C_Normed_Algebra_of_BoundedFunctions X) st f = F & g = G & h = H holds

( H = F * G iff for x being Element of X holds h . x = (f . x) * (g . x) )

let F, G, H be Point of (C_Normed_Algebra_of_BoundedFunctions X); :: thesis: ( f = F & g = G & h = H implies ( H = F * G iff for x being Element of X holds h . x = (f . x) * (g . x) ) )

reconsider f1 = F, g1 = G, h1 = H as VECTOR of (C_Algebra_of_BoundedFunctions X) ;

A1: ( H = F * G iff h1 = f1 * g1 ) ;

assume ( f = F & g = G & h = H ) ; :: thesis: ( H = F * G iff for x being Element of X holds h . x = (f . x) * (g . x) )

hence ( H = F * G iff for x being Element of X holds h . x = (f . x) * (g . x) ) by A1, Th7; :: thesis: verum

for F, G, H being Point of (C_Normed_Algebra_of_BoundedFunctions X) st f = F & g = G & h = H holds

( H = F * G iff for x being Element of X holds h . x = (f . x) * (g . x) )

let f, g, h be Function of X,COMPLEX; :: thesis: for F, G, H being Point of (C_Normed_Algebra_of_BoundedFunctions X) st f = F & g = G & h = H holds

( H = F * G iff for x being Element of X holds h . x = (f . x) * (g . x) )

let F, G, H be Point of (C_Normed_Algebra_of_BoundedFunctions X); :: thesis: ( f = F & g = G & h = H implies ( H = F * G iff for x being Element of X holds h . x = (f . x) * (g . x) ) )

reconsider f1 = F, g1 = G, h1 = H as VECTOR of (C_Algebra_of_BoundedFunctions X) ;

A1: ( H = F * G iff h1 = f1 * g1 ) ;

assume ( f = F & g = G & h = H ) ; :: thesis: ( H = F * G iff for x being Element of X holds h . x = (f . x) * (g . x) )

hence ( H = F * G iff for x being Element of X holds h . x = (f . x) * (g . x) ) by A1, Th7; :: thesis: verum