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

theorem Th22:
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 x0,x1 be Element of REAL m, y0,y1 be Point of REAL-NS m st
     x0 = y0 & x1 = y1 & x0 in X & x1 in X
     holds
       |. f`partial|(X,i)/.x1 - f`partial|(X,i)/.x0 .|
         = ||. (g`partial|(Y,i))/.y1 - (g`partial|(Y,i))/.y0 .||
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 x0,x1 be Element of REAL m, y0,y1 be Point of REAL-NS m;
   assume A2: x0 = y0 & x1 = y1 & x0 in X & x1 in X;
   <*jj*> is Element of REAL 1 by FINSEQ_2:98; then
   reconsider Pt1 = <*jj*> as Point of REAL-NS 1 by REAL_NS1:def 4;
   f`partial|(X,i)/.x1 = partdiff(f,x1,i)
   & f`partial|(X,i)/.x0 = partdiff(f,x0,i) by A2,A1,PDIFF_7:def 5; then
   f`partial|(X,i)/.x1 = partdiff(g,y1,i).Pt1
   & f`partial|(X,i)/.x0 = partdiff(g,y0,i).Pt1 by Th21,A1,A2; then
   f`partial|(X,i)/.x1 - f`partial|(X,i)/.x0
    = partdiff(g,y1,i).Pt1 - partdiff(g,y0,i).Pt1 by REAL_NS1:5; then
A3:f`partial|(X,i)/.x1 - f`partial|(X,i)/.x0
    = (partdiff(g,y1,i) - partdiff(g,y0,i)).Pt1 by LOPBAN_1:40;
   ||.Pt1.|| = |.1.| by PDIFF_8:2; then
   ||.Pt1.|| = 1 by ABSVALUE:def 1; then
A4: ||. (partdiff(g,y1,i) - partdiff(g,y0,i)).Pt1 .||
   = ||. partdiff(g,y1,i) - partdiff(g,y0,i) .|| * 1 by Th20;
   g is_partial_differentiable_on Y,i by A1,PDIFF_7:33; then
   g`partial|(Y,i)/.y1 = partdiff(g,y1,i) &
   g`partial|(Y,i)/.y0 = partdiff(g,y0,i) by A1,A2,PDIFF_1:def 20;
   hence thesis by A3,A4,REAL_NS1:1;
end;
