let m be non zero Element of NAT ; 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 ; 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); 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)); 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; 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) )
reconsider f1 = F, g1 = G as VECTOR of (RAlgebra 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 X holds g . x = a * (f . x)
; G = a * F
then
g1 = a * f1
by A1, FUNCSDOM:4;
hence
G = a * F
by C0SP1:8; verum