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

theorem Th23:
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
 holds
   (f is_partial_differentiable_on X,i & f`partial|(X,i) is_continuous_on X)
     iff
   (g is_partial_differentiable_on Y,i & g`partial|(Y,i) is_continuous_on Y)
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;
   hereby assume
A2: f is_partial_differentiable_on X,i & f`partial|(X,i) is_continuous_on X;
    hence g is_partial_differentiable_on Y,i by A1,PDIFF_7:33; then
A3: dom (g`partial|(Y,i)) = Y by PDIFF_1:def 20;
    for y0 be Point of REAL-NS m, r be Real st y0 in Y & 0 < r
    ex s be Real st 0 < s & for y1 be Point of REAL-NS m st
      y1 in Y & ||. y1- y0 .|| < s holds
        ||. (g`partial|(Y,i))/.y1 - (g`partial|(Y,i))/.y0 .|| < r
    proof
     let y0 be Point of REAL-NS m, r be Real;
     reconsider x0 = y0 as Element of REAL m by REAL_NS1:def 4;
     assume
A4:   y0 in Y & 0 < r; then
     consider s be Real such that
A5:   0 < s & for x1 be Element of REAL m st x1 in X & |. x1-x0 .| <s holds
       |. f`partial|(X,i)/.x1 - f`partial|(X,i)/.x0 .| < r
         by A1,A2,PDIFF_7:38;
     take s;
     thus 0 < s by A5;
     let y1 be Point of REAL-NS m;
     reconsider x1 = y1 as Element of REAL m by REAL_NS1:def 4;
     assume
A6:   y1 in Y & ||. y1- y0 .|| < s; then
A7:  |. x1- x0 .| < s by REAL_NS1:1,5;
     |. f`partial|(X,i)/.x1 - f`partial|(X,i)/.x0 .|
       = ||. (g`partial|(Y,i))/.y1 - (g`partial|(Y,i))/.y0 .||
          by A4,A6,A1,A2,Th22;
     hence ||. (g`partial|(Y,i))/.y1 - (g`partial|(Y,i))/.y0 .|| < r
       by A7,A5,A6,A1;
    end;
    hence g`partial|(Y,i) is_continuous_on Y by A3,NFCONT_1:19;
   end;
   assume
A8: g is_partial_differentiable_on Y,i
  & g`partial|(Y,i) is_continuous_on Y; then
A9:f is_partial_differentiable_on X,i by A1,PDIFF_7:33; then
A10:dom (f`partial|(X,i)) = X by PDIFF_7:def 5;
   for x0 be Element of REAL m, r be Real st x0 in X & 0 < r
   ex s be Real st 0 < s & for x1 be Element of REAL m
    st x1 in X & |. x1 - x0 .| < s holds
     |. (f`partial|(X,i))/.x1 - (f`partial|(X,i))/.x0 .| < r
   proof
    let x0 be Element of REAL m, r be Real;
    reconsider y0 = x0 as Point of REAL-NS m by REAL_NS1:def 4;
    assume
A11:  x0 in X & 0 < r; then
    consider s be Real such that
A12:  0 < s
     & for y1 be Point of REAL-NS m st y1 in Y & ||. y1- y0 .|| < s holds
        ||. g`partial|(Y,i)/.y1 - g`partial|(Y,i)/.y0 .|| < r
           by A1,A8,NFCONT_1:19;
    take s;
    thus 0 < s by A12;
    let x1 be Element of REAL m;
    reconsider y1 = x1 as Element of REAL-NS m by REAL_NS1:def 4;
    assume
A13:  x1 in X & |. x1- x0 .| < s;
    |. x1 - x0 .| = ||. y1 - y0 .|| by REAL_NS1:1,5; then
    ||. g`partial|(Y,i)/.y1 - g`partial|(Y,i)/.y0 .|| < r by A12,A13,A1;
    hence
    |. f`partial|(X,i)/.x1 - f`partial|(X,i)/.x0 .| < r by A11,A13,A1,A9,Th22;
   end;
   hence (f is_partial_differentiable_on X,i &
      f`partial|(X,i) is_continuous_on X) by A8,A10,A1,PDIFF_7:33,38;
end;
