let X be non empty set ; :: thesis: for f, g, h being Function of X,REAL
for F, G, H being Point of (R_Normed_Algebra_of_BoundedFunctions X) 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: for F, G, H being Point of (R_Normed_Algebra_of_BoundedFunctions X) 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 Point of (R_Normed_Algebra_of_BoundedFunctions X); :: 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) ) )
reconsider f1 = F, g1 = G, h1 = H as VECTOR of (R_Algebra_of_BoundedFunctions X) ;
A1: ( H = F + G iff h1 = f1 + g1 ) ;
assume ( f = F & g = G & h = H ) ; :: thesis: ( H = F + G iff for x being Element of X holds h . x = (f . x) + (g . x) )
hence ( H = F + G iff for x being Element of X holds h . x = (f . x) + (g . x) ) by A1, Th12; :: thesis: verum