let S be non empty compact TopSpace; :: thesis: for T being NormedLinearTopSpace
for F, G, H being Point of (R_NormSpace_of_ContinuousFunctions (S,T))
for f, g, h being Function of S,T st f = F & g = G & h = H holds
( H = F + G iff for x being Element of S holds h . x = (f . x) + (g . x) )
let T be NormedLinearTopSpace; :: thesis: for F, G, H being Point of (R_NormSpace_of_ContinuousFunctions (S,T))
for f, g, h being Function of S,T st f = F & g = G & h = H holds
( H = F + G iff for x being Element of S holds h . x = (f . x) + (g . x) )
let F, G, H be Point of (R_NormSpace_of_ContinuousFunctions (S,T)); :: thesis: for f, g, h being Function of S,T st f = F & g = G & h = H holds
( H = F + G iff for x being Element of S holds h . x = (f . x) + (g . x) )
let f, g, h be Function of S,T; :: thesis: ( f = F & g = G & h = H implies ( H = F + G iff for x being Element of S holds h . x = (f . x) + (g . x) ) )
reconsider f1 = F, g1 = G, h1 = H as VECTOR of (R_VectorSpace_of_ContinuousFunctions (S,T)) ;
( H = F + G iff h1 = f1 + g1 ) ;
hence ( f = F & g = G & h = H implies ( H = F + G iff for x being Element of S holds h . x = (f . x) + (g . x) ) ) by Th7; :: thesis: verum