let X, Y, Z be RealLinearSpace; :: thesis: for f, g, h being VECTOR of () 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 (); :: 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 ;
reconsider g1 = g as VECTOR of (RealVectSpace ( the carrier of [:X,Y:],Z)) by ;
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)) )
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 ;
[x,y] is Element of [:X,Y:] ;
hence h9 . (x,y) = (f9 . (x,y)) + (g9 . (x,y)) by ; :: thesis: verum
end;
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 )
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)
proof
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;
then h1 = f1 + g1 by LOPBAN_1:1;
hence h = f + g by ; :: thesis: verum
end;
hence ( 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)) ) by A2; :: thesis: verum