let F, G be PartFunc of COMPLEX,COMPLEX; :: thesis: ( dom F = X & ( for x being Complex st x in X holds
F /. x = diff (f,x) ) & dom G = X & ( for x being Complex st x in X holds
G /. x = diff (f,x) ) implies F = G )
assume that
A7: dom F = X and
A8: for x being Complex st x in X holds
F /. x = diff (f,x) and
A9: dom G = X and
A10: for x being Complex st x in X holds
G /. x = diff (f,x) ; :: thesis: F = G

now :: thesis: for x being Element of COMPLEX st x in dom F holds
F /. x = G /. x

let x be Element of COMPLEX ; :: thesis: ( x in dom F implies F /. x = G /. x )
assume A11: x in dom F ; :: thesis: F /. x = G /. x
then F /. x = diff (f,x) by A7, A8;
hence F /. x = G /. x by A7, A10, A11; :: thesis: verum

end;

hence F = G by A7, A9, PARTFUN2:1; :: thesis: verum