reserve i,n,m for Nat;

theorem Th30:
for X be set, m,n be non zero Nat, f be PartFunc of REAL m,REAL n,
    g be PartFunc of REAL-NS m,REAL-NS n st f=g holds
  f is_differentiable_on X iff g is_differentiable_on X
proof
   let X be set,
       m,n be non zero Nat,
       f be PartFunc of REAL m,REAL n,
       g be PartFunc of REAL-NS m,REAL-NS n;
   assume A1: f=g;
A2: now assume A3: f is_differentiable_on X; then
A4: X c=dom f &
    for x be Element of REAL m st x in X holds f|X is_differentiable_in x;
    now let y be Point of REAL-NS m;
     reconsider x=y as Element of REAL m by REAL_NS1:def 4;
     assume y in X; then
     f|X is_differentiable_in x by A3; then
     ex g be PartFunc of REAL-NS m,REAL-NS n, y be Point of REAL-NS m st
       (f|X)=g & x=y & g is_differentiable_in y by PDIFF_1:def 7;
     hence g|X is_differentiable_in y by A1;
    end;
    hence g is_differentiable_on X by A1,A4,NDIFF_1:def 8;
   end;
   now assume A5: g is_differentiable_on X;
    hence X c= dom f by A1,NDIFF_1:def 8;
A6: X c=dom g
    & for x be Point of REAL-NS m st x in X holds g|X is_differentiable_in x
       by A5,NDIFF_1:def 8;
    now let y be Element of REAL m;
     reconsider x=y as Point of REAL-NS m by REAL_NS1:def 4;
     assume y in X; then
     g|X is_differentiable_in x by A5,NDIFF_1:def 8;
     hence f|X is_differentiable_in y by A1,PDIFF_1:def 7;
    end;
    hence f is_differentiable_on X by A1,A6;
   end;
   hence thesis by A2;
end;
