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 st f = F & g = G & h = H holds
( H = F * G iff for x being Element of X holds h . x = (f . x) * (g . 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 st f = F & g = G & h = H holds
( H = F * G iff for x being Element of X holds h . x = (f . x) * (g . 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 st f = F & g = G & h = H holds
( H = F * G iff for x being Element of X holds h . x = (f . x) * (g . 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 st f = F & g = G & h = H holds
( H = F * G iff for x being Element of X holds h . x = (f . x) * (g . x) )
let f, g, h be PartFunc of (REAL m),REAL; ( f = F & g = G & h = H implies ( H = F * G iff for x being Element of X holds h . x = (f . x) * (g . x) ) )
assume A1:
( f = F & g = G & h = H )
; ( H = F * G iff for x being Element of X holds h . x = (f . x) * (g . x) )
reconsider f1 = F, g1 = G, h1 = H as VECTOR of (RAlgebra X) by TARSKI:def 3;
hereby ( ( for x being Element of X holds h . x = (f . x) * (g . x) ) implies H = F * G )
end;
assume
for x being Element of X holds h . x = (f . x) * (g . x)
; H = F * G
then
h1 = f1 * g1
by A1, FUNCSDOM:2;
hence
H = F * G
by C0SP1:8; verum