let X, Y, Z be RealLinearSpace; :: thesis: for f, g, h being VECTOR of (R_VectorSpace_of_BilinearOperators (X,Y,Z)) holds

( h = f + g iff for x being VECTOR of X

for y being VECTOR of Y holds h . (x,y) = (f . (x,y)) + (g . (x,y)) )

let f, g, h be VECTOR of (R_VectorSpace_of_BilinearOperators (X,Y,Z)); :: thesis: ( h = f + g iff for x being VECTOR of X

for y being VECTOR of Y holds h . (x,y) = (f . (x,y)) + (g . (x,y)) )

reconsider f9 = f, g9 = g, h9 = h as BilinearOperator of X,Y,Z by Def6;

A1: R_VectorSpace_of_BilinearOperators (X,Y,Z) is Subspace of RealVectSpace ( the carrier of [:X,Y:],Z) by RSSPACE:11;

then reconsider f1 = f as VECTOR of (RealVectSpace ( the carrier of [:X,Y:],Z)) by RLSUB_1:10;

reconsider h1 = h as VECTOR of (RealVectSpace ( the carrier of [:X,Y:],Z)) by A1, RLSUB_1:10;

reconsider g1 = g as VECTOR of (RealVectSpace ( the carrier of [:X,Y:],Z)) by A1, RLSUB_1:10;

for y being VECTOR of Y holds h . (x,y) = (f . (x,y)) + (g . (x,y)) ) by A2; :: thesis: verum

( h = f + g iff for x being VECTOR of X

for y being VECTOR of Y holds h . (x,y) = (f . (x,y)) + (g . (x,y)) )

let f, g, h be VECTOR of (R_VectorSpace_of_BilinearOperators (X,Y,Z)); :: thesis: ( h = f + g iff for x being VECTOR of X

for y being VECTOR of Y holds h . (x,y) = (f . (x,y)) + (g . (x,y)) )

reconsider f9 = f, g9 = g, h9 = h as BilinearOperator of X,Y,Z by Def6;

A1: R_VectorSpace_of_BilinearOperators (X,Y,Z) is Subspace of RealVectSpace ( the carrier of [:X,Y:],Z) by RSSPACE:11;

then reconsider f1 = f as VECTOR of (RealVectSpace ( the carrier of [:X,Y:],Z)) by RLSUB_1:10;

reconsider h1 = h as VECTOR of (RealVectSpace ( the carrier of [:X,Y:],Z)) by A1, RLSUB_1:10;

reconsider g1 = g as VECTOR of (RealVectSpace ( the carrier of [:X,Y:],Z)) by A1, RLSUB_1:10;

A2: now :: thesis: ( h = f + g implies for x being Element of X

for y being Element of Y holds h9 . (x,y) = (f9 . (x,y)) + (g9 . (x,y)) )

for y being Element of Y holds h9 . (x,y) = (f9 . (x,y)) + (g9 . (x,y)) )

assume A3:
h = f + g
; :: thesis: for x being Element of X

for y being Element of Y holds h9 . (x,y) = (f9 . (x,y)) + (g9 . (x,y))

let x be Element of X; :: thesis: for y being Element of Y holds h9 . (x,y) = (f9 . (x,y)) + (g9 . (x,y))

let y be Element of Y; :: thesis: h9 . (x,y) = (f9 . (x,y)) + (g9 . (x,y))

A4: h1 = f1 + g1 by A1, A3, RLSUB_1:13;

[x,y] is Element of [:X,Y:] ;

hence h9 . (x,y) = (f9 . (x,y)) + (g9 . (x,y)) by A4, LOPBAN_1:1; :: thesis: verum

end;for y being Element of Y holds h9 . (x,y) = (f9 . (x,y)) + (g9 . (x,y))

let x be Element of X; :: thesis: for y being Element of Y holds h9 . (x,y) = (f9 . (x,y)) + (g9 . (x,y))

let y be Element of Y; :: thesis: h9 . (x,y) = (f9 . (x,y)) + (g9 . (x,y))

A4: h1 = f1 + g1 by A1, A3, RLSUB_1:13;

[x,y] is Element of [:X,Y:] ;

hence h9 . (x,y) = (f9 . (x,y)) + (g9 . (x,y)) by A4, LOPBAN_1:1; :: thesis: verum

now :: thesis: ( ( for x being Element of X

for y being Element of Y holds h9 . (x,y) = (f9 . (x,y)) + (g9 . (x,y)) ) implies h = f + g )

hence
( h = f + g iff for x being VECTOR of Xfor y being Element of Y holds h9 . (x,y) = (f9 . (x,y)) + (g9 . (x,y)) ) implies h = f + g )

assume A5:
for x being Element of X

for y being Element of Y holds h9 . (x,y) = (f9 . (x,y)) + (g9 . (x,y)) ; :: thesis: h = f + g

for z being Element of [:X,Y:] holds h9 . z = (f9 . z) + (g9 . z)

hence h = f + g by A1, RLSUB_1:13; :: thesis: verum

end;for y being Element of Y holds h9 . (x,y) = (f9 . (x,y)) + (g9 . (x,y)) ; :: thesis: h = f + g

for z being Element of [:X,Y:] holds h9 . z = (f9 . z) + (g9 . z)

proof

then
h1 = f1 + g1
by LOPBAN_1:1;
let z be Element of [:X,Y:]; :: thesis: h9 . z = (f9 . z) + (g9 . z)

consider x being Point of X, y being Point of Y such that

A6: z = [x,y] by PRVECT_3:9;

thus h9 . z = h9 . (x,y) by A6

.= (f9 . (x,y)) + (g9 . (x,y)) by A5

.= (f9 . z) + (g9 . z) by A6 ; :: thesis: verum

end;consider x being Point of X, y being Point of Y such that

A6: z = [x,y] by PRVECT_3:9;

thus h9 . z = h9 . (x,y) by A6

.= (f9 . (x,y)) + (g9 . (x,y)) by A5

.= (f9 . z) + (g9 . z) by A6 ; :: thesis: verum

hence h = f + g by A1, RLSUB_1:13; :: thesis: verum

for y being VECTOR of Y holds h . (x,y) = (f . (x,y)) + (g . (x,y)) ) by A2; :: thesis: verum