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

theorem Th71:
for m be non zero Element of NAT, Z be Subset of REAL m, i be Nat,
    f be PartFunc of REAL m,REAL st
  Z is open & 1 <= i & i <= m & f is_partial_differentiable_on Z,i
holds
  f`partial|(Z,i) = (f|Z)`partial|(Z,i)
proof
   let m be non zero Element of NAT, Z be Subset of REAL m,
       i be Nat, f be PartFunc of REAL m,REAL;
   assume A1: Z is open & 1 <= i & i <= m
            & f is_partial_differentiable_on Z,i; then
A2:dom(f `partial|(Z,i)) = Z by Def6;
   for x be Element of REAL m st x in Z holds
     (f `partial|(Z,i))/.x = partdiff((f|Z),x,i)
   proof
    let x be Element of REAL m;
    assume A3: x in Z; then
    f is_partial_differentiable_in x,i
  & (f`partial|(Z,i))/.x = partdiff(f,x,i) by A1,Def6,Th60;
    hence (f`partial|(Z,i))/.x = partdiff((f|Z),x,i) by A1,A3,Th69;
   end;
   hence thesis by A1,A2,Def6;
end;
