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

now

A3: X is Subset of REAL by A1, Th15;
let y be set ; :: 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 by TARSKI:def 3;
for y being set st y in dom F holds
y in X by A2;
then dom F c= X by TARSKI:def 3;
hence dom F = X by A4, XBOOLE_0:def 10; :: thesis: for x being Real st x in X holds
F . x = diff (f,x)

now

let x be Real; :: thesis: ( x in X implies F . x = diff (f,x) )
assume x in X ; :: thesis: F . x = diff (f,x)
then x in dom F by A2;
hence F . x = diff (f,x) by A2; :: thesis: verum

end;

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