let X be non empty closed_interval Subset of REAL; :: thesis: for Y being RealNormSpace
for f, g, h being Point of (R_NormSpace_of_ContinuousFunctions (X,Y))
for f9, g9, h9 being continuous PartFunc of REAL,Y st f9 = f & g9 = g & h9 = h & dom f9 = X & dom g9 = X & dom h9 = X holds
( h = f + g iff for x being Element of X holds h9 /. x = (f9 /. x) + (g9 /. x) )
let Y be RealNormSpace; :: thesis: for f, g, h being Point of (R_NormSpace_of_ContinuousFunctions (X,Y))
for f9, g9, h9 being continuous PartFunc of REAL,Y st f9 = f & g9 = g & h9 = h & dom f9 = X & dom g9 = X & dom h9 = X holds
( h = f + g iff for x being Element of X holds h9 /. x = (f9 /. x) + (g9 /. x) )
let f, g, h be Point of (R_NormSpace_of_ContinuousFunctions (X,Y)); :: thesis: for f9, g9, h9 being continuous PartFunc of REAL,Y st f9 = f & g9 = g & h9 = h & dom f9 = X & dom g9 = X & dom h9 = X holds
( h = f + g iff for x being Element of X holds h9 /. x = (f9 /. x) + (g9 /. x) )
reconsider f1 = f, g1 = g, h1 = h as VECTOR of (R_VectorSpace_of_ContinuousFunctions (X,Y)) ;
A1: ( h = f + g iff h1 = f1 + g1 ) ;
let f9, g9, h9 be continuous PartFunc of REAL,Y; :: thesis: ( f9 = f & g9 = g & h9 = h & dom f9 = X & dom g9 = X & dom h9 = X implies ( h = f + g iff for x being Element of X holds h9 /. x = (f9 /. x) + (g9 /. x) ) )
thus ( f9 = f & g9 = g & h9 = h & dom f9 = X & dom g9 = X & dom h9 = X implies ( h = f + g iff for x being Element of X holds h9 /. x = (f9 /. x) + (g9 /. x) ) ) by A1, Th10; :: thesis: verum