let X be non empty set ; for F, G, H being VECTOR of (R_Algebra_of_BoundedFunctions X)
for f, g, h being Function of X,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_BoundedFunctions X); for f, g, h being Function of X,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 Function of X,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;
A2:
R_Algebra_of_BoundedFunctions X is Subalgebra of RAlgebra X
by Th6;
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:1;
hence
H = F + G
by A2, Th8; verum