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

theorem Th24:
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
 st X = Y & X is open & f = g
 holds
  (for i being Nat st 1 <= i & i <= m holds
    f is_partial_differentiable_on X,i & f`partial|(X,i) is_continuous_on X)
 iff
  g is_differentiable_on Y & g`|Y 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;
   assume A1: X = Y & X is open & f = g;
   hereby assume
A3:for i be Nat st 1 <=i & i <= m holds
     f is_partial_differentiable_on X,i &
     f`partial|(X,i) is_continuous_on X;
    now let i be Nat;
     assume A4: 1 <=i & i <= m; then
     f is_partial_differentiable_on X,i &
     f`partial|(X,i) is_continuous_on X by A3;
     hence g is_partial_differentiable_on Y,i &
      g`partial|(Y,i) is_continuous_on Y by A1,A4,Th23;
    end;
    hence g is_differentiable_on Y & g`|Y is_continuous_on Y
      by A1,PDIFF_8:22;
   end;
   assume A5: g is_differentiable_on Y & g`|Y is_continuous_on Y;
   let i be Nat;
   assume A6: 1 <=i & i <= m; then
   g is_partial_differentiable_on Y,i &
   g`partial|(Y,i) is_continuous_on Y by A1,A5,PDIFF_8:22;
   hence thesis by A1,A6,Th23;
end;
