let Z be open Subset of REAL ; :: thesis: for f1, f2 being PartFunc of REAL ,REAL st Z c= dom (f1 (#) f2) & f1 is_differentiable_on Z & f2 is_differentiable_on Z holds
( f1 (#) f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 (#) f2) `| Z) . x = ((f2 . x) * (diff f1,x)) + ((f1 . x) * (diff f2,x)) ) )
let f1, f2 be PartFunc of REAL ,REAL ; :: thesis: ( Z c= dom (f1 (#) f2) & f1 is_differentiable_on Z & f2 is_differentiable_on Z implies ( f1 (#) f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 (#) f2) `| Z) . x = ((f2 . x) * (diff f1,x)) + ((f1 . x) * (diff f2,x)) ) ) )
assume that
A1: Z c= dom (f1 (#) f2) and
A2: f1 is_differentiable_on Z and
A3: f2 is_differentiable_on Z ; :: thesis: ( f1 (#) f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 (#) f2) `| Z) . x = ((f2 . x) * (diff f1,x)) + ((f1 . x) * (diff f2,x)) ) )

now

let x0 be Real; :: thesis: ( x0 in Z implies f1 (#) f2 is_differentiable_in x0 )
assume A4: x0 in Z ; :: thesis: f1 (#) f2 is_differentiable_in x0
then A5: f1 is_differentiable_in x0 by A2, Th16;
f2 is_differentiable_in x0 by A3, A4, Th16;
hence f1 (#) f2 is_differentiable_in x0 by A5, Th24; :: thesis: verum

end;

hence A6: f1 (#) f2 is_differentiable_on Z by A1, Th16; :: thesis: for x being Real st x in Z holds
((f1 (#) f2) `| Z) . x = ((f2 . x) * (diff f1,x)) + ((f1 . x) * (diff f2,x))

now

let x be Real; :: thesis: ( x in Z implies ((f1 (#) f2) `| Z) . x = ((f2 . x) * (diff f1,x)) + ((f1 . x) * (diff f2,x)) )
assume A7: x in Z ; :: thesis: ((f1 (#) f2) `| Z) . x = ((f2 . x) * (diff f1,x)) + ((f1 . x) * (diff f2,x))
then A8: f1 is_differentiable_in x by A2, Th16;
A9: f2 is_differentiable_in x by A3, A7, Th16;
thus ((f1 (#) f2) `| Z) . x = diff (f1 (#) f2),x by A6, A7, Def8
.= ((f2 . x) * (diff f1,x)) + ((f1 . x) * (diff f2,x)) by A8, A9, Th24 ; :: thesis: verum

end;

hence for x being Real st x in Z holds
((f1 (#) f2) `| Z) . x = ((f2 . x) * (diff f1,x)) + ((f1 . x) * (diff f2,x)) ; :: thesis: verum