let X be non empty set ; :: thesis: for Y being ComplexNormSpace
for f, h being VECTOR of (C_VectorSpace_of_BoundedFunctions X,Y)
for f', h' being bounded Function of X,the carrier of Y st f' = f & h' = h holds
for c being Complex holds
( h = c * f iff for x being Element of X holds h' . x = c * (f' . x) )
let Y be ComplexNormSpace; :: thesis: for f, h being VECTOR of (C_VectorSpace_of_BoundedFunctions X,Y)
for f', h' being bounded Function of X,the carrier of Y st f' = f & h' = h holds
for c being Complex holds
( h = c * f iff for x being Element of X holds h' . x = c * (f' . x) )
let f, h be VECTOR of (C_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 c being Complex holds
( h = c * f iff for x being Element of X holds h' . x = c * (f' . x) )
let f', h' be bounded Function of X,the carrier of Y; :: thesis: ( f' = f & h' = h implies for c being Complex holds
( h = c * f iff for x being Element of X holds h' . x = c * (f' . x) ) )
assume that
A1: f' = f and
A2: h' = h ; :: thesis: for c being Complex holds
( h = c * f iff for x being Element of X holds h' . x = c * (f' . x) )
let c be Complex; :: thesis: ( h = c * f iff for x being Element of X holds h' . x = c * (f' . x) )
A3: C_VectorSpace_of_BoundedFunctions X,Y is Subspace of ComplexVectSpace X,Y by Th8, CSSPACE:13;
then reconsider f1 = f, h1 = h as VECTOR of (ComplexVectSpace X,Y) by CLVECT_1:30;

A4: now

assume A5: h = c * f ; :: thesis: for x being Element of X holds h' . x = c * (f' . x)
let x be Element of X; :: thesis: h' . x = c * (f' . x)
h1 = c * f1 by A3, A5, CLVECT_1:34;
hence h' . x = c * (f' . x) by A1, A2, CLOPBAN1:13; :: thesis: verum

end;

now

assume for x being Element of X holds h' . x = c * (f' . x) ; :: thesis: h = c * f
then h1 = c * f1 by A1, A2, CLOPBAN1:13;
hence h = c * f by A3, CLVECT_1:34; :: thesis: verum

end;

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