
theorem Th1:
  for m be non zero Element of NAT,
      Z be set,I be non empty FinSequence of NAT,
      f be PartFunc of REAL m,REAL st f is_partial_differentiable_on Z,I
        holds dom (f`partial|(Z,I)) = Z
proof
  let m be non zero Element of NAT, Z be set,
      I be non empty FinSequence of NAT,
      f be PartFunc of REAL m,REAL;
  assume
A1: f is_partial_differentiable_on Z,I;
  reconsider k=(len I)-1 as Element of NAT by INT_1:5,FINSEQ_1:20;
A2: f`partial|(Z,I) = ((PartDiffSeq(f,Z,I)).k)`partial|(Z,I/.(k+1))
                                                  by PDIFF_9:def 7;
  (PartDiffSeq(f,Z,I)).k is_partial_differentiable_on Z,I/.(k+1)
                                                  by A1;
  hence thesis by A2,PDIFF_9:def 6;
end;
