defpred S₁[ Element of REAL 3] means $1 in D;
deffunc H₁( Element of REAL 3) -> Element of REAL = partdiff f,$1,1;
consider F being PartFunc of (REAL 3),REAL such that
A2: ( ( for u being Element of REAL 3 holds
( u in dom F iff S₁[u] ) ) & ( for u being Element of REAL 3 st u in dom F holds
F . u = H₁(u) ) ) from SEQ_1:sch 3();
take F ; :: thesis: ( dom F = D & ( for u being Element of REAL 3 st u in D holds
F . u = partdiff f,u,1 ) )
for y being set st y in dom F holds
y in D by A2;
then A3: dom F c= D by TARSKI:def 3;

now

let y be set ; :: thesis: ( y in D implies y in dom F )
assume A4: y in D ; :: thesis: y in dom F
D c= dom f by A1, Def16;
then D is Subset of (REAL 3) by XBOOLE_1:1;
hence y in dom F by A2, A4; :: thesis: verum

end;

then D c= dom F by TARSKI:def 3;
hence dom F = D by A3, XBOOLE_0:def 10; :: thesis: for u being Element of REAL 3 st u in D holds
F . u = partdiff f,u,1
let u be Element of REAL 3; :: thesis: ( u in D implies F . u = partdiff f,u,1 )
assume u in D ; :: thesis: F . u = partdiff f,u,1
then u in dom F by A2;
hence F . u = partdiff f,u,1 by A2; :: thesis: verum