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

theorem
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 be Nat st 1 <= i & i <= m holds
     f is_partial_differentiable_on X,i & f`partial|(X,i) is_continuous_on X )
 iff
  f is_differentiable_on X & 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; then
    g is_differentiable_on Y & g`|Y is_continuous_on Y by A1,PDIFF_8:22;
    hence f is_differentiable_on X & g`|Y is_continuous_on Y by A1,PDIFF_6:30;
   end;
   assume f is_differentiable_on X & g`|Y is_continuous_on Y; then
A5:g is_differentiable_on Y & g`|Y is_continuous_on Y by A1,PDIFF_6:30;
   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;
