reserve m,n for non zero Element of NAT;
reserve i,j,k for Element of NAT;
reserve Z for set;

theorem Th72:
for f be PartFunc of REAL m,REAL, I be non empty FinSequence of NAT
  st f is_partial_differentiable_on Z,I
holds (f`partial|(Z,I)) |Z = f`partial|(Z,I)
proof
   let f be PartFunc of REAL m,REAL, I be non empty FinSequence of NAT;
   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 Def7; then
   dom((PartDiffSeq(f,Z,I)).(k+1)) = Z by A1,Def6;
   hence (f`partial|(Z,I)) |Z = f`partial|(Z,I) by RELAT_1:68;
end;
