let X be non empty set ; :: thesis: 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); :: thesis: 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 ; :: 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) )
A4: R_Algebra_of_BoundedFunctions X is Subalgebra of RAlgebra X by Th03;
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 A6: 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 A4, A6, RLSUB121;
hence h . x = (f . x) + (g . x) by A1, FUNCSDOM:10; :: 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:10;
hence H = F + G by A4, RLSUB121; :: thesis: verum