let X be non empty TopSpace; for F, G, H being VECTOR of (R_Algebra_of_ContinuousFunctions X)
for f, g, h being RealMap of X
for a being 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, H be VECTOR of (R_Algebra_of_ContinuousFunctions X); for f, g, h being RealMap of X
for a being 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, h be RealMap of X; for a being 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; ( 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 )
; ( G = a * F iff for x being Element of X holds g . x = a * (f . x) )
A2:
R_Algebra_of_ContinuousFunctions X is Subalgebra of RAlgebra the carrier of X
by C0SP1:6;
reconsider f1 = F, g1 = G as VECTOR of (RAlgebra the carrier of X) by TARSKI:def 3;
hereby ( ( for x being Element of X holds g . x = a * (f . x) ) implies G = a * F )
end;
assume
for x being Element of the carrier of X holds g . x = a * (f . x)
; G = a * F
then
g1 = a * f1
by A1, FUNCSDOM:4;
hence
G = a * F
by A2, C0SP1:8; verum