let X be non empty set ; :: thesis: for Y being ComplexNormSpace

for f, g, h being Point of (C_NormSpace_of_BoundedFunctions (X,Y))

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 Point of (C_NormSpace_of_BoundedFunctions (X,Y))

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 Point of (C_NormSpace_of_BoundedFunctions (X,Y)); :: 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) )

reconsider f1 = f, g1 = g, h1 = h as VECTOR of (C_VectorSpace_of_BoundedFunctions (X,Y)) ;

A1: ( h = f + g iff h1 = f1 + g1 ) ;

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 ( f9 = f & g9 = g & h9 = h ) ; :: thesis: ( h = f + g iff for x being Element of X holds h9 . x = (f9 . x) + (g9 . x) )

hence ( h = f + g iff for x being Element of X holds h9 . x = (f9 . x) + (g9 . x) ) by A1, Th9; :: thesis: verum

for f, g, h being Point of (C_NormSpace_of_BoundedFunctions (X,Y))

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 Point of (C_NormSpace_of_BoundedFunctions (X,Y))

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 Point of (C_NormSpace_of_BoundedFunctions (X,Y)); :: 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) )

reconsider f1 = f, g1 = g, h1 = h as VECTOR of (C_VectorSpace_of_BoundedFunctions (X,Y)) ;

A1: ( h = f + g iff h1 = f1 + g1 ) ;

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 ( f9 = f & g9 = g & h9 = h ) ; :: thesis: ( h = f + g iff for x being Element of X holds h9 . x = (f9 . x) + (g9 . x) )

hence ( h = f + g iff for x being Element of X holds h9 . x = (f9 . x) + (g9 . x) ) by A1, Th9; :: thesis: verum