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

now

let x0 be Real; :: thesis: ( x0 in Z implies - f1 is_differentiable_in x0 )
assume x0 in Z ; :: thesis: - f1 is_differentiable_in x0
then f1 is_differentiable_in x0 by A2, FDIFF_1:16;
hence - f1 is_differentiable_in x0 by Th22; :: thesis: verum

end;

hence A3: - f1 is_differentiable_on Z by A1, FDIFF_1:16; :: thesis: for x being Real st x in Z holds
((- f1) `| Z) . x = - (diff f1,x)

now

let x be Real; :: thesis: ( x in Z implies ((- f1) `| Z) . x = - (diff f1,x) )
assume A4: x in Z ; :: thesis: ((- f1) `| Z) . x = - (diff f1,x)
then A5: f1 is_differentiable_in x by A2, FDIFF_1:16;
thus ((- f1) `| Z) . x = diff (- f1),x by A3, A4, FDIFF_1:def 8
.= - (diff f1,x) by A5, Th22 ; :: thesis: verum

end;

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