let a be Real; :: thesis: for X being non empty closed_interval Subset of REAL
for Y being RealNormSpace
for f, h being Point of (R_NormSpace_of_ContinuousFunctions (X,Y))
for f9, h9 being continuous PartFunc of REAL,Y st f9 = f & h9 = h & dom f9 = X & dom h9 = X holds
( h = a * f iff for x being Element of X holds h9 /. x = a * (f9 /. x) )
let X be non empty closed_interval Subset of REAL; :: thesis: for Y being RealNormSpace
for f, h being Point of (R_NormSpace_of_ContinuousFunctions (X,Y))
for f9, h9 being continuous PartFunc of REAL,Y st f9 = f & h9 = h & dom f9 = X & dom h9 = X 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_ContinuousFunctions (X,Y))
for f9, h9 being continuous PartFunc of REAL,Y st f9 = f & h9 = h & dom f9 = X & dom h9 = X 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_ContinuousFunctions (X,Y)); :: thesis: for f9, h9 being continuous PartFunc of REAL,Y st f9 = f & h9 = h & dom f9 = X & dom h9 = X holds
( h = a * f iff for x being Element of X holds h9 /. x = a * (f9 /. x) )
reconsider f1 = f, h1 = h as VECTOR of (R_VectorSpace_of_ContinuousFunctions (X,Y)) ;
let f9, h9 be continuous PartFunc of REAL,Y; :: thesis: ( f9 = f & h9 = h & dom f9 = X & dom h9 = X implies ( h = a * f iff for x being Element of X holds h9 /. x = a * (f9 /. x) ) )
assume A1: ( f9 = f & h9 = h & dom f9 = X & dom h9 = X ) ; :: 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, Th11; :: thesis: verum