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