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

for f, g, h being VECTOR of (C_VectorSpace_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 VECTOR of (C_VectorSpace_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 VECTOR of (C_VectorSpace_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) )

A1: C_VectorSpace_of_BoundedFunctions (X,Y) is Subspace of ComplexVectSpace (X,Y) by Th7, CSSPACE:11;

then reconsider f1 = f as VECTOR of (ComplexVectSpace (X,Y)) by CLVECT_1:29;

reconsider h1 = h as VECTOR of (ComplexVectSpace (X,Y)) by A1, CLVECT_1:29;

reconsider g1 = g as VECTOR of (ComplexVectSpace (X,Y)) by A1, CLVECT_1:29;

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) )

for f, g, h being VECTOR of (C_VectorSpace_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 VECTOR of (C_VectorSpace_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 VECTOR of (C_VectorSpace_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) )

A1: C_VectorSpace_of_BoundedFunctions (X,Y) is Subspace of ComplexVectSpace (X,Y) by Th7, CSSPACE:11;

then reconsider f1 = f as VECTOR of (ComplexVectSpace (X,Y)) by CLVECT_1:29;

reconsider h1 = h as VECTOR of (ComplexVectSpace (X,Y)) by A1, CLVECT_1:29;

reconsider g1 = g as VECTOR of (ComplexVectSpace (X,Y)) by A1, CLVECT_1:29;

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 A1, A4, CLVECT_1:32;

hence h9 . x = (f9 . x) + (g9 . x) by A2, CLOPBAN1:11; :: thesis: verum

end;let x be Element of X; :: thesis: h9 . x = (f9 . x) + (g9 . x)

h1 = f1 + g1 by A1, A4, CLVECT_1:32;

hence h9 . x = (f9 . x) + (g9 . x) by A2, CLOPBAN1:11; :: thesis: verum

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

hence
( h = f + g iff for x being Element of X holds h9 . x = (f9 . x) + (g9 . x) )
by A3; :: thesis: verumassume
for x being Element of X holds h9 . x = (f9 . x) + (g9 . x)
; :: thesis: h = f + g

then h1 = f1 + g1 by A2, CLOPBAN1:11;

hence h = f + g by A1, CLVECT_1:32; :: thesis: verum

end;then h1 = f1 + g1 by A2, CLOPBAN1:11;

hence h = f + g by A1, CLVECT_1:32; :: thesis: verum