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

reconsider f1 = f as VECTOR of (R_VectorSpace_of_BoundedBilinearOperators (X,Y,Z)) ;

reconsider h1 = h as VECTOR of (R_VectorSpace_of_BoundedBilinearOperators (X,Y,Z)) ;

( h = a * f iff h1 = a * f1 ) ;

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

for y being VECTOR of Y holds h . (x,y) = a * (f . (x,y)) ) by Th25; :: 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 Point of (R_NormSpace_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)) )

reconsider f1 = f as VECTOR of (R_VectorSpace_of_BoundedBilinearOperators (X,Y,Z)) ;

reconsider h1 = h as VECTOR of (R_VectorSpace_of_BoundedBilinearOperators (X,Y,Z)) ;

( h = a * f iff h1 = a * f1 ) ;

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

for y being VECTOR of Y holds h . (x,y) = a * (f . (x,y)) ) by Th25; :: thesis: verum