let X, Y be RealNormSpace; :: thesis: for f, h being Point of (R_NormSpace_of_BoundedLinearOperators (X,Y))
for a being Real holds
( h = a * f iff for x being VECTOR of X holds h . x = a * (f . x) )
let f, h be Point of (R_NormSpace_of_BoundedLinearOperators (X,Y)); :: thesis: for a being Real holds
( h = a * f iff for x being VECTOR of X holds h . x = a * (f . x) )
let a be Real; :: thesis: ( h = a * f iff for x being VECTOR of X holds h . x = a * (f . x) )
reconsider f1 = f as VECTOR of (R_VectorSpace_of_BoundedLinearOperators (X,Y)) ;
reconsider h1 = h as VECTOR of (R_VectorSpace_of_BoundedLinearOperators (X,Y)) ;
( h = a * f iff h1 = a * f1 ) ;
hence ( h = a * f iff for x being VECTOR of X holds h . x = a * (f . x) ) by Th25; :: thesis: verum