let X be non empty set ; 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; 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)); 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; ( 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 )
; 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; ( h = c * f iff for x being Element of X holds h9 . x = c * (f9 . x) )
A2:
now ( h1 = c * f1 implies h = c * f )end;
now ( h = c * f implies h1 = c * f1 )end;
hence
( h = c * f iff for x being Element of X holds h9 . x = c * (f9 . x) )
by A1, A2, Th10; verum