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

theorem
for f be PartFunc of REAL m,REAL, I,G be non empty FinSequence of NAT
 st f is_partial_differentiable_on Z,G holds
  (f`partial|(Z,G))`partial|(Z,I) = f`partial|(Z,(G^I))
proof
  let f be PartFunc of REAL m,REAL, I,G be non empty FinSequence of NAT;
  assume AS: f is_partial_differentiable_on Z,G;
  thus (f`partial|(Z,G))`partial|(Z,I)
     = (PartDiffSeq(f,Z,(G^I))).(len G + len I) by Th73,AS
    .= f`partial|(Z,(G^I)) by FINSEQ_1:22;
end;
