let S, T be RealNormSpace; :: thesis: for f, h being VECTOR of (R_VectorSpace_of_ContinuousFunctions (S,T))
for a being Real holds
( h = a * f iff for x being Element of S holds h . x = a * (f . x) )
let f, h be VECTOR of (R_VectorSpace_of_ContinuousFunctions (S,T)); :: thesis: for a being Real holds
( h = a * f iff for x being Element of S holds h . x = a * (f . x) )
set G = { f where f is Function of the carrier of S, the carrier of T : f is_continuous_on the carrier of S } ;
A1: ( f in { f where f is Function of the carrier of S, the carrier of T : f is_continuous_on the carrier of S } & h in { f where f is Function of the carrier of S, the carrier of T : f is_continuous_on the carrier of S } ) ;
then ex x being Function of the carrier of S, the carrier of T st
( f = x & x is_continuous_on the carrier of S ) ;
then reconsider f9 = f as Function of S,T ;
ex x being Function of the carrier of S, the carrier of T st
( h = x & x is_continuous_on the carrier of S ) by A1;
then reconsider h9 = h as Function of S,T ;
let a be Real; :: thesis: ( h = a * f iff for x being Element of S holds h . x = a * (f . x) )
A2: R_VectorSpace_of_ContinuousFunctions (S,T) is Subspace of RealVectSpace ( the carrier of S,T) by Th11, RSSPACE:11;
then reconsider f1 = f as VECTOR of (RealVectSpace ( the carrier of S,T)) by RLSUB_1:10;
reconsider h1 = h as VECTOR of (RealVectSpace ( the carrier of S,T)) by A2, RLSUB_1:10;

A3: now :: thesis: ( h = a * f implies for x being Element of S holds h9 . x = a * (f9 . x) )

assume A4: h = a * f ; :: thesis: for x being Element of S holds h9 . x = a * (f9 . x)
let x be Element of S; :: thesis: h9 . x = a * (f9 . x)
h1 = a * f1 by A2, A4, RLSUB_1:14;
hence h9 . x = a * (f9 . x) by LOPBAN_1:2; :: thesis: verum

end;

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

assume for x being Element of S holds h9 . x = a * (f9 . x) ; :: thesis: h = a * f
then h1 = a * f1 by LOPBAN_1:2;
hence h = a * f by A2, RLSUB_1:14; :: thesis: verum

end;

hence ( h = a * f iff for x being Element of S holds h . x = a * (f . x) ) by A3; :: thesis: verum