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

theorem Th21:
for f be PartFunc of REAL m,REAL n, g be PartFunc of REAL-NS m,REAL-NS n,
    X be Subset of REAL m, Y be Subset of REAL-NS m, i being Nat st
  1 <= i & i <= m & X is open & g = f & X = Y &
  f is_partial_differentiable_on X,i
holds
   for x be Element of REAL m, y be Point of REAL-NS m st x in X & x = y
     holds partdiff(f,x,i) = partdiff(g,y,i).<*1*>
proof
   let f be PartFunc of REAL m,REAL n, g be PartFunc of REAL-NS m,REAL-NS n,
       X be Subset of REAL m, Y be Subset of REAL-NS m, i be Nat;
   assume
A1: 1 <= i & i <= m & X is open & g = f & X = Y
     & f is_partial_differentiable_on X,i;
   let x be Element of REAL m, y be Point of REAL-NS m;
   assume A2: x in X & x = y; then
   f is_partial_differentiable_in x,i by A1,PDIFF_7:34; then
   ex g0 be PartFunc of REAL-NS m,REAL-NS n, y0 be Point of REAL-NS m st
    f = g0 & x = y0 & partdiff(f,x,i) = partdiff(g0,y0,i).<*1*>
      by PDIFF_1:def 14;
   hence partdiff(f,x,i) = partdiff(g,y,i).<*1*> by A1,A2;
end;
