let X, Y, Z be RealNormSpace; :: thesis: for f, g, h being VECTOR of (R_VectorSpace_of_BoundedBilinearOperators (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_BoundedBilinearOperators (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)) )

A1: R_VectorSpace_of_BoundedBilinearOperators (X,Y,Z) is Subspace of R_VectorSpace_of_BilinearOperators (X,Y,Z) by RSSPACE:11;

then reconsider f1 = f as VECTOR of (R_VectorSpace_of_BilinearOperators (X,Y,Z)) by RLSUB_1:10;

reconsider h1 = h as VECTOR of (R_VectorSpace_of_BilinearOperators (X,Y,Z)) by A1, RLSUB_1:10;

reconsider g1 = g as VECTOR of (R_VectorSpace_of_BilinearOperators (X,Y,Z)) by A1, RLSUB_1:10;

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

then h1 = f1 + g1 by Th16;

hence h = f + g by A1, RLSUB_1:13; :: 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_BoundedBilinearOperators (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)) )

A1: R_VectorSpace_of_BoundedBilinearOperators (X,Y,Z) is Subspace of R_VectorSpace_of_BilinearOperators (X,Y,Z) by RSSPACE:11;

then reconsider f1 = f as VECTOR of (R_VectorSpace_of_BilinearOperators (X,Y,Z)) by RLSUB_1:10;

reconsider h1 = h as VECTOR of (R_VectorSpace_of_BilinearOperators (X,Y,Z)) by A1, RLSUB_1:10;

reconsider g1 = g as VECTOR of (R_VectorSpace_of_BilinearOperators (X,Y,Z)) by A1, RLSUB_1:10;

hereby :: thesis: ( ( for x being VECTOR of X

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

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

assume A2:
h = f + g
; :: thesis: for x being Element of X

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

let x be Element of X; :: thesis: for y being Element of Y holds h . (x,y) = (f . (x,y)) + (g . (x,y))

let y be Element of Y; :: thesis: h . (x,y) = (f . (x,y)) + (g . (x,y))

h1 = f1 + g1 by A1, A2, RLSUB_1:13;

hence h . (x,y) = (f . (x,y)) + (g . (x,y)) by Th16; :: thesis: verum

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

let x be Element of X; :: thesis: for y being Element of Y holds h . (x,y) = (f . (x,y)) + (g . (x,y))

let y be Element of Y; :: thesis: h . (x,y) = (f . (x,y)) + (g . (x,y))

h1 = f1 + g1 by A1, A2, RLSUB_1:13;

hence h . (x,y) = (f . (x,y)) + (g . (x,y)) by Th16; :: thesis: verum

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

then h1 = f1 + g1 by Th16;

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