reserve i,n,m for Nat;

theorem Th2:
for n,m be non zero Nat, f be PartFunc of REAL m,REAL n,
    g be PartFunc of REAL-NS m,REAL-NS n,
    x be Element of REAL m, y be Point of REAL-NS m st
  f=g & x=y holds f is_differentiable_in x iff g is_differentiable_in y
proof
   let n,m be non zero Nat,
       f be PartFunc of REAL m,REAL n,
       g be PartFunc of REAL-NS m,REAL-NS n,
       x be Element of REAL m, y be Point of REAL-NS m;
   assume A1: f=g & x=y;
   hereby assume f is_differentiable_in x; then
    ex g be PartFunc of REAL-NS m,REAL-NS n, y be Point of REAL-NS m st
     f=g & x=y & g is_differentiable_in y by PDIFF_1:def 7;
    hence g is_differentiable_in y by A1;
   end;
   assume g is_differentiable_in y;
   hence f is_differentiable_in x by A1,PDIFF_1:def 7;
end;
