deffunc H₁( Element of COMPLEX ) -> Element of COMPLEX = diff (f,$1);
defpred S₁[ set ] means $1 in X;
consider F being PartFunc of COMPLEX,COMPLEX such that
A2: ( ( for x being Element of COMPLEX holds
( x in dom F iff S₁[x] ) ) & ( for x being Element of COMPLEX st x in dom F holds
F . x = H₁(x) ) ) from SEQ_1:sch 3();
take F ; :: thesis: ( dom F = X & ( for x being Complex st x in X holds
F /. x = diff (f,x) ) )

now :: thesis: for y being object st y in X holds
y in dom F

A3: X is Subset of COMPLEX by A1, Th8;
let y be object ; :: thesis: ( y in X implies y in dom F )
assume y in X ; :: thesis: y in dom F
hence y in dom F by A2, A3; :: thesis: verum

end;

then A4: X c= dom F ;
for y being object st y in dom F holds
y in X by A2;
then dom F c= X ;
hence dom F = X by A4; :: thesis: for x being Complex st x in X holds
F /. x = diff (f,x)

now :: thesis: for x being Complex st x in X holds
F /. x = diff (f,x)

let x be Complex; :: thesis: ( x in X implies F /. x = diff (f,x) )
assume A5: x in X ; :: thesis: F /. x = diff (f,x)
x in COMPLEX by XCMPLX_0:def 2;
then A6: x in dom F by A2, A5;
then F . x = diff (f,x) by A2;
hence F /. x = diff (f,x) by A6, PARTFUN1:def 6; :: thesis: verum

end;

hence for x being Complex st x in X holds
F /. x = diff (f,x) ; :: thesis: verum