let X be non empty set ; :: thesis: for a being Complex

for f, g being Function of X,COMPLEX

for F, G being Point of (C_Normed_Algebra_of_BoundedFunctions X) 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 Function of X,COMPLEX

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

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

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

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

hence ( G = a * F iff for x being Element of X holds g . x = a * (f . x) ) by A1, Th6; :: thesis: verum

for f, g being Function of X,COMPLEX

for F, G being Point of (C_Normed_Algebra_of_BoundedFunctions X) 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 Function of X,COMPLEX

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

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

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

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

hence ( G = a * F iff for x being Element of X holds g . x = a * (f . x) ) by A1, Th6; :: thesis: verum