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

theorem Th81:
for f be PartFunc of REAL m,REAL holds
 f is_partial_differentiable_on Z,<*i*> iff f is_partial_differentiable_on Z,i
proof
   let f be PartFunc of REAL m,REAL;
   set I = <*i*>;
A1:len I = 1 by FINSEQ_1:39;
A2:(PartDiffSeq(f,Z,I)).0 = f|Z  by Def7;
   1 in Seg 1; then
A3:1 in dom I by FINSEQ_1:38;
   I/.(Z0 + 1) =I.1 by A3,PARTFUN1:def 6; then
   I/.(Z0 + 1) = i;
   hence f is_partial_differentiable_on Z,I implies
   f is_partial_differentiable_on Z,i by A2,A1;
   assume A4:f is_partial_differentiable_on Z,i;
   now let k be Element of NAT;
    assume k <= len I - 1; then
A5: k = 0 by A1; then
    I/.(k+1) = I.1 by A3,PARTFUN1:def 6; then
    I/.(k+1) = i;
    hence (PartDiffSeq(f,Z,I)).k is_partial_differentiable_on Z,I/.(k+1)
       by A4,A5,Def7;
   end;
   hence thesis;
end;
