let X be non empty set ; :: thesis: for Y being ComplexNormSpace
for f, h being Point of ()
for f9, h9 being bounded Function of X, the carrier of Y st f9 = f & h9 = h holds
for c being Complex holds
( h = c * f iff for x being Element of X holds h9 . x = c * (f9 . x) )

let Y be ComplexNormSpace; :: thesis: for f, h being Point of ()
for f9, h9 being bounded Function of X, the carrier of Y st f9 = f & h9 = h holds
for c being Complex holds
( h = c * f iff for x being Element of X holds h9 . x = c * (f9 . x) )

let f, h be Point of (); :: thesis: for f9, h9 being bounded Function of X, the carrier of Y st f9 = f & h9 = h holds
for c being Complex holds
( h = c * f iff for x being Element of X holds h9 . x = c * (f9 . x) )

let f9, h9 be bounded Function of X, the carrier of Y; :: thesis: ( f9 = f & h9 = h implies for c being Complex holds
( h = c * f iff for x being Element of X holds h9 . x = c * (f9 . x) ) )

assume A1: ( f9 = f & h9 = h ) ; :: thesis: for c being Complex holds
( h = c * f iff for x being Element of X holds h9 . x = c * (f9 . x) )

reconsider h1 = h as VECTOR of ;
reconsider f1 = f as VECTOR of ;
let c be Complex; :: thesis: ( h = c * f iff for x being Element of X holds h9 . x = c * (f9 . x) )
A2: now :: thesis: ( h1 = c * f1 implies h = c * f )
assume h1 = c * f1 ; :: thesis: h = c * f
hence h = (Mult_ ((),(ComplexVectSpace (X,Y)))) . [c,f1] by CLVECT_1:def 1
.= c * f by CLVECT_1:def 1 ;
:: thesis: verum
end;
now :: thesis: ( h = c * f implies h1 = c * f1 )
assume h = c * f ; :: thesis: h1 = c * f1
hence h1 = (Mult_ ((),(ComplexVectSpace (X,Y)))) . [c,f] by CLVECT_1:def 1
.= c * f1 by CLVECT_1:def 1 ;
:: thesis: verum
end;
hence ( h = c * f iff for x being Element of X holds h9 . x = c * (f9 . x) ) by A1, A2, Th10; :: thesis: verum