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) )
reconsider f9 = f, h9 = h as LinearOperator of X,Y by Def6;
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_LinearOperators (X,Y) is Subspace of RealVectSpace ( the carrier of X,Y) by Th14, RSSPACE:11;
then reconsider f1 = f as VECTOR of (RealVectSpace ( the carrier of X,Y)) by RLSUB_1:10;
reconsider h1 = h as VECTOR of (RealVectSpace ( the carrier of X,Y)) by A1, RLSUB_1:10;

A2: now :: thesis: ( h = a * f implies for x being Element of X holds h9 . x = a * (f9 . x) )

assume A3: h = a * f ; :: thesis: for x being Element of X holds h9 . x = a * (f9 . x)
let x be Element of X; :: thesis: h9 . x = a * (f9 . x)
h1 = a * f1 by A1, A3, RLSUB_1:14;
hence h9 . x = a * (f9 . x) by Th2; :: thesis: verum

end;

now :: thesis: ( ( for x being Element of X holds h9 . x = a * (f9 . x) ) implies h = a * f )

assume for x being Element of X holds h9 . x = a * (f9 . x) ; :: thesis: h = a * f
then h1 = a * f1 by Th2;
hence h = a * f by A1, RLSUB_1:14; :: thesis: verum

end;

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