set W = BoundedBilinearOperators (X,Y,Z);
A1: for v, u being VECTOR of (R_VectorSpace_of_BilinearOperators (X,Y,Z)) st v in BoundedBilinearOperators (X,Y,Z) & u in BoundedBilinearOperators (X,Y,Z) holds
v + u in BoundedBilinearOperators (X,Y,Z)

proof

let v, u be VECTOR of (R_VectorSpace_of_BilinearOperators (X,Y,Z)); :: thesis: ( v in BoundedBilinearOperators (X,Y,Z) & u in BoundedBilinearOperators (X,Y,Z) implies v + u in BoundedBilinearOperators (X,Y,Z) )
assume that
A2: v in BoundedBilinearOperators (X,Y,Z) and
A3: u in BoundedBilinearOperators (X,Y,Z) ; :: thesis: v + u in BoundedBilinearOperators (X,Y,Z)
reconsider f = v + u as BilinearOperator of X,Y,Z by EQSET;
f is Lipschitzian

proof

reconsider v1 = v as Lipschitzian BilinearOperator of X,Y,Z by A2, Def9;
consider K2 being Real such that
A4: 0 <= K2 and
A5: for x being VECTOR of X
for y being VECTOR of Y holds ||.(v1 . (x,y)).|| <= (K2 * ||.x.||) * ||.y.|| by Def8;
reconsider u1 = u as Lipschitzian BilinearOperator of X,Y,Z by A3, Def9;
consider K1 being Real such that
A6: 0 <= K1 and
A7: for x being VECTOR of X
for y being VECTOR of Y holds ||.(u1 . (x,y)).|| <= (K1 * ||.x.||) * ||.y.|| by Def8;
take K3 = K1 + K2; :: according to LOPBAN_9:def 3 :: thesis: ( 0 <= K3 & ( for x being VECTOR of X
for y being VECTOR of Y holds ||.(f . (x,y)).|| <= (K3 * ||.x.||) * ||.y.|| ) )

now :: thesis: for x being VECTOR of X
for y being VECTOR of Y holds ||.(f . (x,y)).|| <= (K3 * ||.x.||) * ||.y.||

let x be VECTOR of X; :: thesis: for y being VECTOR of Y holds ||.(f . (x,y)).|| <= (K3 * ||.x.||) * ||.y.||
let y be VECTOR of Y; :: thesis: ||.(f . (x,y)).|| <= (K3 * ||.x.||) * ||.y.||
A8: ||.((u1 . (x,y)) + (v1 . (x,y))).|| <= ||.(u1 . (x,y)).|| + ||.(v1 . (x,y)).|| by NORMSP_1:def 1;
A9: ||.(v1 . (x,y)).|| <= (K2 * ||.x.||) * ||.y.|| by A5;
||.(u1 . (x,y)).|| <= (K1 * ||.x.||) * ||.y.|| by A7;
then A10: ||.(u1 . (x,y)).|| + ||.(v1 . (x,y)).|| <= ((K1 * ||.x.||) * ||.y.||) + ((K2 * ||.x.||) * ||.y.||) by A9, XREAL_1:7;
||.(f . (x,y)).|| = ||.((u1 . (x,y)) + (v1 . (x,y))).|| by Th16;
hence ||.(f . (x,y)).|| <= (K3 * ||.x.||) * ||.y.|| by A8, A10, XXREAL_0:2; :: thesis: verum

end;

hence ( 0 <= K3 & ( for x being VECTOR of X
for y being VECTOR of Y holds ||.(f . (x,y)).|| <= (K3 * ||.x.||) * ||.y.|| ) ) by A4, A6; :: thesis: verum

end;

hence v + u in BoundedBilinearOperators (X,Y,Z) by Def9; :: thesis: verum

end;

for a being Real
for v being VECTOR of (R_VectorSpace_of_BilinearOperators (X,Y,Z)) st v in BoundedBilinearOperators (X,Y,Z) holds
a * v in BoundedBilinearOperators (X,Y,Z)

proof

let a be Real; :: thesis: for v being VECTOR of (R_VectorSpace_of_BilinearOperators (X,Y,Z)) st v in BoundedBilinearOperators (X,Y,Z) holds
a * v in BoundedBilinearOperators (X,Y,Z)
let v be VECTOR of (R_VectorSpace_of_BilinearOperators (X,Y,Z)); :: thesis: ( v in BoundedBilinearOperators (X,Y,Z) implies a * v in BoundedBilinearOperators (X,Y,Z) )
assume A11: v in BoundedBilinearOperators (X,Y,Z) ; :: thesis: a * v in BoundedBilinearOperators (X,Y,Z)
reconsider f = a * v as BilinearOperator of X,Y,Z by EQSET;
f is Lipschitzian

proof

reconsider v1 = v as Lipschitzian BilinearOperator of X,Y,Z by A11, Def9;
consider K being Real such that
A12: 0 <= K and
A13: for x being VECTOR of X
for y being VECTOR of Y holds ||.(v1 . (x,y)).|| <= (K * ||.x.||) * ||.y.|| by Def8;
take |.a.| * K ; :: according to LOPBAN_9:def 3 :: thesis: ( 0 <= |.a.| * K & ( for x being VECTOR of X
for y being VECTOR of Y holds ||.(f . (x,y)).|| <= ((|.a.| * K) * ||.x.||) * ||.y.|| ) )

A14: now :: thesis: for x being VECTOR of X
for y being VECTOR of Y holds ||.(f . (x,y)).|| <= ((|.a.| * K) * ||.x.||) * ||.y.||

let x be VECTOR of X; :: thesis: for y being VECTOR of Y holds ||.(f . (x,y)).|| <= ((|.a.| * K) * ||.x.||) * ||.y.||
let y be VECTOR of Y; :: thesis: ||.(f . (x,y)).|| <= ((|.a.| * K) * ||.x.||) * ||.y.||
0 <= |.a.| by COMPLEX1:46;
then A15: |.a.| * ||.(v1 . (x,y)).|| <= |.a.| * ((K * ||.x.||) * ||.y.||) by A13, XREAL_1:64;
||.(a * (v1 . (x,y))).|| = |.a.| * ||.(v1 . (x,y)).|| by NORMSP_1:def 1;
hence ||.(f . (x,y)).|| <= ((|.a.| * K) * ||.x.||) * ||.y.|| by A15, Th17; :: thesis: verum

end;

0 <= |.a.| by COMPLEX1:46;
hence ( 0 <= |.a.| * K & ( for x being VECTOR of X
for y being VECTOR of Y holds ||.(f . (x,y)).|| <= ((|.a.| * K) * ||.x.||) * ||.y.|| ) ) by A12, A14; :: thesis: verum

end;

hence a * v in BoundedBilinearOperators (X,Y,Z) by Def9; :: thesis: verum

end;

hence BoundedBilinearOperators (X,Y,Z) is linearly-closed by A1, RLSUB_1:def 1; :: thesis: verum