deffunc H₁( Element of REAL 2) -> Real = hpartdiff22 f,$1;
defpred S₁[ Element of REAL 2] means $1 in Z;
consider F being PartFunc of (REAL 2),REAL such that
A2: ( ( for z being Element of REAL 2 holds
( z in dom F iff S₁[z] ) ) & ( for z being Element of REAL 2 st z in dom F holds
F . z = H₁(z) ) ) from SEQ_1:sch 3();
take F ; :: thesis: ( dom F = Z & ( for z being Element of REAL 2 st z in Z holds
F . z = hpartdiff22 f,z ) )

now

Z c= dom f by A1, Def13;
then A3: Z is Subset of (REAL 2) by XBOOLE_1:1;
let y be set ; :: thesis: ( y in Z implies y in dom F )
assume y in Z ; :: thesis: y in dom F
hence y in dom F by A2, A3; :: thesis: verum

end;

then A4: Z c= dom F by TARSKI:def 3;
for y being set st y in dom F holds
y in Z by A2;
then dom F c= Z by TARSKI:def 3;
hence dom F = Z by A4, XBOOLE_0:def 10; :: thesis: for z being Element of REAL 2 st z in Z holds
F . z = hpartdiff22 f,z

hereby :: thesis: verum

let z be Element of REAL 2; :: thesis: ( z in Z implies F . z = hpartdiff22 f,z )
assume z in Z ; :: thesis: F . z = hpartdiff22 f,z
then z in dom F by A2;
hence F . z = hpartdiff22 f,z by A2; :: thesis: verum

end;