let X, Y be RealLinearSpace; :: thesis: for f, h being VECTOR of (R_VectorSpace_of_LinearOperators 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_LinearOperators 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 f' = f, h' = h as LinearOperator of X,Y by Def7;
A1: R_VectorSpace_of_LinearOperators X,Y is Subspace of RealVectSpace the carrier of X,Y by Th17, RSSPACE:13;
then reconsider f1 = f as VECTOR of (RealVectSpace the carrier of X,Y) by RLSUB_1:18;
reconsider h1 = h as VECTOR of (RealVectSpace the carrier of X,Y) by A1, RLSUB_1:18;

A2: now

assume A3: 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, A3, RLSUB_1:22;
hence h' . x = a * (f' . x) by Th5; :: thesis: verum

end;

now

assume for x being Element of X holds h' . x = a * (f' . x) ; :: thesis: h = a * f
then h1 = a * f1 by Th5;
hence h = a * f by A1, RLSUB_1:22; :: thesis: verum

end;

hence ( h = a * f iff for x being VECTOR of X holds h . x = a * (f . x) ) by A2; :: thesis: verum