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
A6: dom F = X and
A7: for x being Complex st x in X holds
F /. x = diff f,x and
A8: dom G = X and
A9: for x being Complex st x in X holds
G /. x = diff f,x ; :: thesis: F = G

now

let x be Complex; :: thesis: ( x in dom F implies F /. x = G /. x )
assume A10: x in dom F ; :: thesis: F /. x = G /. x
then F /. x = diff f,x by A6, A7;
hence F /. x = G /. x by A6, A9, A10; :: thesis: verum

end;

hence F = G by A6, A8, PARTFUN2:3; :: thesis: verum