let X, Y be RealNormSpace; :: thesis: for f, h being VECTOR of (R_VectorSpace_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 VECTOR of (R_VectorSpace_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) )
A1: R_VectorSpace_of_BoundedLinearOperators (X,Y) is Subspace of R_VectorSpace_of_LinearOperators (X,Y) by Th26, RSSPACE:13;
then reconsider f1 = f as VECTOR of (R_VectorSpace_of_LinearOperators (X,Y)) by RLSUB_1:18;
reconsider h1 = h as VECTOR of (R_VectorSpace_of_LinearOperators (X,Y)) by A1, RLSUB_1:18;

hereby :: thesis: ( ( for x being VECTOR of X holds h . x = a * (f . x) ) implies h = a * f )

assume A2: h = a * f ; :: thesis: for x being Element of X holds h . x = a * (f . x)
let x be Element of X; :: thesis: h . x = a * (f . x)
h1 = a * f1 by A1, A2, RLSUB_1:22;
hence h . x = a * (f . x) by Th21; :: thesis: verum

end;

assume for x being Element of X holds h . x = a * (f . x) ; :: thesis: h = a * f
then h1 = a * f1 by Th21;
hence h = a * f by A1, RLSUB_1:22; :: thesis: verum