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;
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:1;
hence H = F + G by C0SP1:8; :: thesis: verum