let X be non empty set ; :: thesis: for Y being RealNormSpace
for f, h being VECTOR of (R_VectorSpace_of_BoundedFunctions X,Y)
for f9, h9 being bounded Function of X,the carrier of Y st f9 = f & h9 = h holds
for a being Real holds
( h = a * f iff for x being Element of X holds h9 . x = a * (f9 . x) )
let Y be RealNormSpace; :: thesis: for f, h being VECTOR of (R_VectorSpace_of_BoundedFunctions X,Y)
for f9, h9 being bounded Function of X,the carrier of Y st f9 = f & h9 = h holds
for a being Real holds
( h = a * f iff for x being Element of X holds h9 . x = a * (f9 . x) )
let f, h be VECTOR of (R_VectorSpace_of_BoundedFunctions X,Y); :: thesis: for f9, h9 being bounded Function of X,the carrier of Y st f9 = f & h9 = h holds
for a being Real holds
( h = a * f iff for x being Element of X holds h9 . x = a * (f9 . x) )
let f9, h9 be bounded Function of X,the carrier of Y; :: thesis: ( f9 = f & h9 = h implies for a being Real holds
( h = a * f iff for x being Element of X holds h9 . x = a * (f9 . x) ) )
assume A1: ( f9 = f & h9 = h ) ; :: thesis: for a being Real holds
( h = a * f iff for x being Element of X holds h9 . x = a * (f9 . x) )
let a be Real; :: thesis: ( h = a * f iff for x being Element of X holds h9 . x = a * (f9 . x) )
A2: 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;

A3: now

assume A4: 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 A2, A4, RLSUB_1:22;
hence h9 . x = a * (f9 . x) by A1, LOPBAN_1:15; :: thesis: verum

end;

now

assume for x being Element of X holds h9 . x = a * (f9 . x) ; :: thesis: h = a * f
then h1 = a * f1 by A1, LOPBAN_1:15;
hence h = a * f by A2, RLSUB_1:22; :: thesis: verum

end;

hence ( h = a * f iff for x being Element of X holds h9 . x = a * (f9 . x) ) by A3; :: thesis: verum