let m be non zero Element of NAT ; :: thesis: for k being Element of NAT
for X being non empty open Subset of (REAL m)
for F, G, H being VECTOR of (R_Algebra_of_Ck_Functions (k,X))
for f, g, h being PartFunc of (REAL m),REAL
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 k be Element of NAT ; :: thesis: for X being non empty open Subset of (REAL m)
for F, G, H being VECTOR of (R_Algebra_of_Ck_Functions (k,X))
for f, g, h being PartFunc of (REAL m),REAL
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 X be non empty open Subset of (REAL m); :: thesis: for F, G, H being VECTOR of (R_Algebra_of_Ck_Functions (k,X))
for f, g, h being PartFunc of (REAL m),REAL
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_Ck_Functions (k,X)); :: thesis: for f, g, h being PartFunc of (REAL m),REAL
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 PartFunc of (REAL m),REAL; :: thesis: 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; :: 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) )
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 A2: 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 A2, C0SP1:8;
hence g . x = a * (f . x) by A1, FUNCSDOM:4; :: 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:4;
hence G = a * F by C0SP1:8; :: thesis: verum