let X be non empty set ; :: thesis: for Y being RealNormSpace
for f, h being VECTOR of (R_VectorSpace_of_BoundedFunctions X,Y)
for f', h' being bounded Function of X,the carrier of Y st f' = f & h' = h holds
for a being Real holds
( h = a * f iff for x being Element of X holds h' . x = a * (f' . x) )
let Y be RealNormSpace; :: thesis: for f, h being VECTOR of (R_VectorSpace_of_BoundedFunctions X,Y)
for f', h' being bounded Function of X,the carrier of Y st f' = f & h' = h holds
for a being Real holds
( h = a * f iff for x being Element of X holds h' . x = a * (f' . x) )
let f, h be VECTOR of (R_VectorSpace_of_BoundedFunctions X,Y); :: thesis: for f', h' being bounded Function of X,the carrier of Y st f' = f & h' = h holds
for a being Real holds
( h = a * f iff for x being Element of X holds h' . x = a * (f' . x) )
let f', h' be bounded Function of X,the carrier of Y; :: thesis: ( f' = f & h' = h implies for a being Real holds
( h = a * f iff for x being Element of X holds h' . x = a * (f' . x) ) )
assume that
A1: f' = f and
A2: h' = h ; :: thesis: for a being Real holds
( h = a * f iff for x being Element of X holds h' . x = a * (f' . x) )
let a be Real; :: thesis: ( h = a * f iff for x being Element of X holds h' . x = a * (f' . x) )
A3: R_VectorSpace_of_BoundedFunctions X,Y is Subspace of RealVectSpace X,Y by Th7, RSSPACE:13;
then reconsider f1 = f, h1 = h as VECTOR of (RealVectSpace X,Y) by RLSUB_1:18;

A4: now

assume A5: 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 A3, A5, RLSUB_1:22;
hence h' . x = a * (f' . x) by A1, A2, LOPBAN_1:15; :: 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 A1, A2, LOPBAN_1:15;
hence h = a * f by A3, RLSUB_1:22; :: thesis: verum

end;

hence ( h = a * f iff for x being Element of X holds h' . x = a * (f' . x) ) by A4; :: thesis: verum