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

for a being Real holds

( h = a * f iff for x being VECTOR of X

for y being VECTOR of Y holds h . (x,y) = a * (f . (x,y)) )

let f, h be VECTOR of (R_VectorSpace_of_BoundedBilinearOperators (X,Y,Z)); :: thesis: for a being Real holds

( h = a * f iff for x being VECTOR of X

for y being VECTOR of Y holds h . (x,y) = a * (f . (x,y)) )

let a be Real; :: thesis: ( h = a * f iff for x being VECTOR of X

for y being VECTOR of Y holds h . (x,y) = a * (f . (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;

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

then h1 = a * f1 by Th17;

hence h = a * f by A1, RLSUB_1:14; :: thesis: verum

for a being Real holds

( h = a * f iff for x being VECTOR of X

for y being VECTOR of Y holds h . (x,y) = a * (f . (x,y)) )

let f, h be VECTOR of (R_VectorSpace_of_BoundedBilinearOperators (X,Y,Z)); :: thesis: for a being Real holds

( h = a * f iff for x being VECTOR of X

for y being VECTOR of Y holds h . (x,y) = a * (f . (x,y)) )

let a be Real; :: thesis: ( h = a * f iff for x being VECTOR of X

for y being VECTOR of Y holds h . (x,y) = a * (f . (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;

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

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

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

assume A2:
h = a * f
; :: thesis: for x being Element of X

for y being Element of Y holds h . (x,y) = a * (f . (x,y))

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

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

h1 = a * f1 by A1, A2, RLSUB_1:14;

hence h . (x,y) = a * (f . (x,y)) by Th17; :: thesis: verum

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

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

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

h1 = a * f1 by A1, A2, RLSUB_1:14;

hence h . (x,y) = a * (f . (x,y)) by Th17; :: thesis: verum

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

then h1 = a * f1 by Th17;

hence h = a * f by A1, RLSUB_1:14; :: thesis: verum