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 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 ; :: 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 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); :: 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 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)); :: thesis: 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; :: thesis: ( 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 ) ; :: thesis: ( 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 :: thesis: ( ( for x being Element of X holds h . x = (f . x) * (g . x) ) implies H = F * G )

assume A2: H = F * G ; :: thesis: for x being Element of X holds h . x = (f . x) * (g . x)
let x be Element of X; :: thesis: h . x = (f . x) * (g . x)
h1 = f1 * g1 by A2, C0SP1:8;
hence h . x = (f . x) * (g . x) by A1, FUNCSDOM:2; :: thesis: verum

end;

assume for x being Element of X holds h . x = (f . x) * (g . x) ; :: thesis: H = F * G
then h1 = f1 * g1 by A1, FUNCSDOM:2;
hence H = F * G by C0SP1:8; :: thesis: verum