reserve m, n for non zero Element of NAT;
reserve i, j, k for Element of NAT;
reserve Z for Subset of REAL 2;
reserve c for Real;
reserve I for non empty FinSequence of NAT;
reserve d1, d2 for Element of REAL;

theorem DOMP1:
  for m being non zero Element of NAT,
      X being Subset of (REAL m),
      I being non empty FinSequence of NAT,
      f being PartFunc of (REAL m), REAL st
      f is_partial_differentiable_on X, I
    holds dom(f`partial| (X,I)) = X
  proof
    let m be non zero Element of NAT,
    Z be Subset of (REAL m),
    I be non empty FinSequence of NAT,
    f be PartFunc of (REAL m), REAL;
    reconsider k=(len I)-1 as Element of NAT by INT_1:5,FINSEQ_1:20;
    assume f is_partial_differentiable_on Z, I; then
    A1:(PartDiffSeq(f,Z,I)).k is_partial_differentiable_on Z,I/.(k+1);
    dom((PartDiffSeq(f,Z,I)).(k+1))
    = dom(((PartDiffSeq(f,Z,I)).k)`partial|(Z,I/.(k+1))) by PDIFF_9:def 7;
    hence thesis by A1, PDIFF_9:def 6;
  end;
