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

theorem Th82:
for f be PartFunc of REAL m,REAL, Z be Subset of REAL m,
    i be Element of NAT
 st Z is open & 1 <= i & i <= m & f is_partial_differentiable_on Z,i
holds f`partial|(Z,<*i*>) = f`partial|(Z,i)
proof
   let f be PartFunc of REAL m,REAL, Z be Subset of REAL m,
       i be Element of NAT;
   assume A1:Z is open & 1 <= i & i <= m & f is_partial_differentiable_on Z,i;
   set I = <*i*>;
A2:PartDiffSeq(f,Z,I).0 = f|Z by Def7;
   1 in Seg 1; then
   1 in dom I by FINSEQ_1:38; then
   I/.(0 + 1) = I.1 by PARTFUN1:def 6; then
A3:I/.(0 + 1) = i;
   thus f`partial|(Z,I) = (PartDiffSeq(f,Z,I)).1 by FINSEQ_1:39
                  .= (PartDiffSeq(f,Z,I).0)`partial|(Z,I/.(0 + 1)) by Def7
                  .= f`partial|(Z,i) by A2,A3,A1,Th71;
end;
