let X be non empty set ; :: thesis: for a being Real
for F, G being VECTOR of (R_Algebra_of_BoundedFunctions X)
for f, g being Function of X,REAL 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; :: thesis: for F, G being VECTOR of (R_Algebra_of_BoundedFunctions X)
for f, g being Function of X,REAL 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 (R_Algebra_of_BoundedFunctions X); :: thesis: for f, g being Function of X,REAL 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 ; :: 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) )
A3: R_Algebra_of_BoundedFunctions X is Subalgebra of RAlgebra X by Th03;
reconsider f1 = F, g1 = G as VECTOR of (RAlgebra X) by TARSKI:def 3;

hereby :: thesis: ( ( for x being Element of X holds g . x = a * (f . x) ) implies G = a * F )

assume A5: G = a * F ; :: thesis: for x being Element of X holds g . x = a * (f . x)
let x be Element of X; :: thesis: g . x = a * (f . x)
g1 = a * f1 by A3, A5, RLSUB121;
hence g . x = a * (f . x) by A1, FUNCSDOM:15; :: thesis: verum

end;

assume for x being Element of X holds g . x = a * (f . x) ; :: thesis: G = a * F
then g1 = a * f1 by A1, FUNCSDOM:15;
hence G = a * F by A3, RLSUB121; :: thesis: verum