let X be non empty set ; :: thesis: for Y being RealNormSpace
for f, h being Point of (R_NormSpace_of_BoundedFunctions (X,Y))
for f9, h9 being bounded Function of X, the carrier of Y st f9 = f & h9 = h holds
for a being Real holds
( h = a * f iff for x being Element of X holds h9 . x = a * (f9 . x) )
let Y be RealNormSpace; :: thesis: for f, h being Point of (R_NormSpace_of_BoundedFunctions (X,Y))
for f9, h9 being bounded Function of X, the carrier of Y st f9 = f & h9 = h holds
for a being Real holds
( h = a * f iff for x being Element of X holds h9 . x = a * (f9 . x) )
let f, h be Point of (R_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 a being Real holds
( h = a * f iff for x being Element of X holds h9 . x = a * (f9 . x) )
let f9, h9 be bounded Function of X, the carrier of Y; :: thesis: ( f9 = f & h9 = h implies for a being Real holds
( h = a * f iff for x being Element of X holds h9 . x = a * (f9 . x) ) )
assume A1: ( f9 = f & h9 = h ) ; :: thesis: for a being Real holds
( h = a * f iff for x being Element of X holds h9 . x = a * (f9 . x) )
reconsider h1 = h as VECTOR of (R_VectorSpace_of_BoundedFunctions (X,Y)) ;
reconsider f1 = f as VECTOR of (R_VectorSpace_of_BoundedFunctions (X,Y)) ;
let a be Real; :: thesis: ( h = a * f iff for x being Element of X holds h9 . x = a * (f9 . x) )
( h = a * f iff h1 = a * f1 ) ;
hence ( h = a * f iff for x being Element of X holds h9 . x = a * (f9 . x) ) by A1, Th9; :: thesis: verum