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 & 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 x0 in Z ; :: thesis: f1 (#) f2 is_differentiable_in x0
then ( f1 is_differentiable_in x0 & f2 is_differentiable_in x0 ) by A2, Th16;
hence f1 (#) f2 is_differentiable_in x0 by Th24; :: thesis: verum

end;

hence A3: 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 A4: x in Z ; :: thesis: ((f1 (#) f2) `| Z) . x = ((f2 . x) * (diff (f1,x))) + ((f1 . x) * (diff (f2,x)))
then A5: ( f1 is_differentiable_in x & f2 is_differentiable_in x ) by A2, Th16;
thus ((f1 (#) f2) `| Z) . x = diff ((f1 (#) f2),x) by A3, A4, Def8
.= ((f2 . x) * (diff (f1,x))) + ((f1 . x) * (diff (f2,x))) by A5, 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