let X be non empty set ; for a being Real
for f, g being Function of X,REAL
for F, G being Point of (R_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 Real; for f, g being Function of X,REAL
for F, G being Point of (R_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,REAL; for F, G being Point of (R_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 (R_Normed_Algebra_of_BoundedFunctions X); ( 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 (R_Algebra_of_BoundedFunctions X) ;
A1:
( G = a * F iff g1 = a * f1 )
;
assume
( f = F & g = G )
; ( 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, Th13; verum