let S, T be RealNormSpace; :: thesis: for Z being Subset of S st Z is open holds
for r being Real
for f being PartFunc of S,T st Z c= dom (r (#) f) & f is_differentiable_on Z holds
( r (#) f is_differentiable_on Z & ( for x being Point of S st x in Z holds
((r (#) f) `| Z) /. x = r * (diff (f,x)) ) )

let Z be Subset of S; :: thesis: ( Z is open implies for r being Real
for f being PartFunc of S,T st Z c= dom (r (#) f) & f is_differentiable_on Z holds
( r (#) f is_differentiable_on Z & ( for x being Point of S st x in Z holds
((r (#) f) `| Z) /. x = r * (diff (f,x)) ) ) )

assume A1: Z is open ; :: thesis: for r being Real
for f being PartFunc of S,T st Z c= dom (r (#) f) & f is_differentiable_on Z holds
( r (#) f is_differentiable_on Z & ( for x being Point of S st x in Z holds
((r (#) f) `| Z) /. x = r * (diff (f,x)) ) )

let r be Real; :: thesis: for f being PartFunc of S,T st Z c= dom (r (#) f) & f is_differentiable_on Z holds
( r (#) f is_differentiable_on Z & ( for x being Point of S st x in Z holds
((r (#) f) `| Z) /. x = r * (diff (f,x)) ) )

let f be PartFunc of S,T; :: thesis: ( Z c= dom (r (#) f) & f is_differentiable_on Z implies ( r (#) f is_differentiable_on Z & ( for x being Point of S st x in Z holds
((r (#) f) `| Z) /. x = r * (diff (f,x)) ) ) )

assume that
A2: Z c= dom (r (#) f) and
A3: f is_differentiable_on Z ; :: thesis: ( r (#) f is_differentiable_on Z & ( for x being Point of S st x in Z holds
((r (#) f) `| Z) /. x = r * (diff (f,x)) ) )

now :: thesis: for x0 being Point of S st x0 in Z holds
r (#) f is_differentiable_in x0
end;
hence A4: r (#) f is_differentiable_on Z by A1, A2, Th31; :: thesis: for x being Point of S st x in Z holds
((r (#) f) `| Z) /. x = r * (diff (f,x))

now :: thesis: for x being Point of S st x in Z holds
((r (#) f) `| Z) /. x = r * (diff (f,x))
let x be Point of S; :: thesis: ( x in Z implies ((r (#) f) `| Z) /. x = r * (diff (f,x)) )
assume A5: x in Z ; :: thesis: ((r (#) f) `| Z) /. x = r * (diff (f,x))
then A6: f is_differentiable_in x by A1, A3, Th31;
thus ((r (#) f) `| Z) /. x = diff ((r (#) f),x) by A4, A5, Def9
.= r * (diff (f,x)) by ; :: thesis: verum
end;
hence for x being Point of S st x in Z holds
((r (#) f) `| Z) /. x = r * (diff (f,x)) ; :: thesis: verum