let X be non empty set ; :: thesis: for Y being ComplexNormSpace
for f, g, h being VECTOR of
for f9, g9, h9 being bounded Function of X, the carrier of Y st f9 = f & g9 = g & h9 = h holds
( h = f + g iff for x being Element of X holds h9 . x = (f9 . x) + (g9 . x) )

let Y be ComplexNormSpace; :: thesis: for f, g, h being VECTOR of
for f9, g9, h9 being bounded Function of X, the carrier of Y st f9 = f & g9 = g & h9 = h holds
( h = f + g iff for x being Element of X holds h9 . x = (f9 . x) + (g9 . x) )

let f, g, h be VECTOR of ; :: thesis: for f9, g9, h9 being bounded Function of X, the carrier of Y st f9 = f & g9 = g & h9 = h holds
( h = f + g iff for x being Element of X holds h9 . x = (f9 . x) + (g9 . x) )

A1: C_VectorSpace_of_BoundedFunctions (X,Y) is Subspace of ComplexVectSpace (X,Y) by ;
then reconsider f1 = f as VECTOR of (ComplexVectSpace (X,Y)) by CLVECT_1:29;
reconsider h1 = h as VECTOR of (ComplexVectSpace (X,Y)) by ;
reconsider g1 = g as VECTOR of (ComplexVectSpace (X,Y)) by ;
let f9, g9, h9 be bounded Function of X, the carrier of Y; :: thesis: ( f9 = f & g9 = g & h9 = h implies ( h = f + g iff for x being Element of X holds h9 . x = (f9 . x) + (g9 . x) ) )
assume A2: ( f9 = f & g9 = g & h9 = h ) ; :: thesis: ( h = f + g iff for x being Element of X holds h9 . x = (f9 . x) + (g9 . x) )
A3: now :: thesis: ( h = f + g implies for x being Element of X holds h9 . x = (f9 . x) + (g9 . x) )
assume A4: h = f + g ; :: thesis: for x being Element of X holds h9 . x = (f9 . x) + (g9 . x)
let x be Element of X; :: thesis: h9 . x = (f9 . x) + (g9 . x)
h1 = f1 + g1 by ;
hence h9 . x = (f9 . x) + (g9 . x) by ; :: thesis: verum
end;
now :: thesis: ( ( for x being Element of X holds h9 . x = (f9 . x) + (g9 . x) ) implies h = f + g )
assume for x being Element of X holds h9 . x = (f9 . x) + (g9 . x) ; :: thesis: h = f + g
then h1 = f1 + g1 by ;
hence h = f + g by ; :: thesis: verum
end;
hence ( h = f + g iff for x being Element of X holds h9 . x = (f9 . x) + (g9 . x) ) by A3; :: thesis: verum