let X be non empty set ; :: thesis: for Y being ComplexNormSpace
for f, h being Point of (C_NormSpace_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 Point of (C_NormSpace_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 Point of (C_NormSpace_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) )
reconsider f1 = f as VECTOR of (C_VectorSpace_of_BoundedFunctions X,Y) ;
reconsider h1 = h as VECTOR of (C_VectorSpace_of_BoundedFunctions X,Y) ;
( h = c * f iff h1 = c * f1 )

proof

A3: now

assume h = c * f ; :: thesis: h1 = c * f1
hence h1 = (Mult_ (ComplexBoundedFunctions X,Y),(ComplexVectSpace X,Y)) . [c,f] by CLVECT_1:def 1
.= c * f1 by CLVECT_1:def 1 ;
:: thesis: verum

end;

now

assume h1 = c * f1 ; :: thesis: h = c * f
hence h = (Mult_ (ComplexBoundedFunctions X,Y),(ComplexVectSpace X,Y)) . [c,f1] by CLVECT_1:def 1
.= c * f by CLVECT_1:def 1 ;
:: thesis: verum

end;

hence ( h = c * f iff h1 = c * f1 ) by A3; :: thesis: verum

end;

hence ( h = c * f iff for x being Element of X holds h' . x = c * (f' . x) ) by A1, A2, Th12; :: thesis: verum