let X be non empty set ; :: thesis: for Y being ComplexNormSpace

for f, h being Point of (C_NormSpace_of_BoundedFunctions (X,Y))

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 (C_NormSpace_of_BoundedFunctions (X,Y))

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 (C_NormSpace_of_BoundedFunctions (X,Y)); :: 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 (C_VectorSpace_of_BoundedFunctions (X,Y)) ;

reconsider f1 = f as VECTOR of (C_VectorSpace_of_BoundedFunctions (X,Y)) ;

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

for f, h being Point of (C_NormSpace_of_BoundedFunctions (X,Y))

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 (C_NormSpace_of_BoundedFunctions (X,Y))

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 (C_NormSpace_of_BoundedFunctions (X,Y)); :: 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 (C_VectorSpace_of_BoundedFunctions (X,Y)) ;

reconsider f1 = f as VECTOR of (C_VectorSpace_of_BoundedFunctions (X,Y)) ;

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_ ((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 = (Mult_ ((ComplexBoundedFunctions (X,Y)),(ComplexVectSpace (X,Y)))) . [c,f1] by CLVECT_1:def 1

.= c * f by CLVECT_1:def 1 ;

:: thesis: verum

now :: thesis: ( h = c * f implies h1 = c * f1 )

hence
( h = c * f iff for x being Element of X holds h9 . x = c * (f9 . x) )
by A1, A2, Th10; :: thesis: verumassume
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;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