
theorem Th16:
  for m be non zero Element of NAT, i be Element of NAT,
      f be PartFunc of REAL m,REAL, X be non empty Subset of REAL m,
      d be Real st X is open & f = X --> d & 1 <= i & i <= m
        holds
      f`partial|(X,i) = X --> 0
proof
  let m be non zero Element of NAT, i be Element of NAT,
  f be PartFunc of REAL m,REAL, X be non empty Subset of REAL m,
  d be Real;
  assume
A1: X is open & f = X --> d & 1 <= i & i <= m;
A2:f is_partial_differentiable_on X,i by A1,Th15;
A3:dom (f`partial|(X,i)) = X by A2,PDIFF_9:def 6;
  now let x be object;
    assume
A4:  x in dom (f`partial|(X,i));
    then
      reconsider x1=x as Element of REAL m;
A5: f is_differentiable_in x1 & diff(f,x1) = REAL m --> 0 by Th12,A1,A3,A4;
A6:  (reproj(i,0*m).1) in REAL m by XREAL_0:def 1,FUNCT_2:5;
A7: partdiff(f,x1,i) = diff(f,x1).(reproj(i,0*m).1) by PDIFF_7:23,A5,A1
                    .= 0 by A6,FUNCOP_1:7,A5;
    thus (f`partial|(X,i)).x =(f`partial|(X,i))/.x1 by A4,PARTFUN1:def 6
                         .= 0 by A7,A2,A4,A3,PDIFF_9:def 6;
  end;
  hence thesis by A3,FUNCOP_1:11;
end;
