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;

then h1 = f1 + g1 by A1, FUNCSDOM:1;

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

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
for x being Element of X holds h . x = (f . x) + (g . x)
; :: thesis: H = F + Gassume 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:1; :: thesis: verum

end;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:1; :: thesis: verum

then h1 = f1 + g1 by A1, FUNCSDOM:1;

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