reserve n for Nat,
        p,p1,p2 for Point of TOP-REAL n,
        x for Real;
reserve n,m for non zero Nat;
reserve i,j for Nat;
reserve f for PartFunc of REAL-NS m,REAL-NS n;
reserve g for PartFunc of REAL m,REAL n;
reserve h for PartFunc of REAL m,REAL;
reserve x for Point of REAL-NS m;
reserve y for Element of REAL m;
reserve X for set;

theorem Th33:
for m,n be non zero Nat, i be Nat,
    f be PartFunc of REAL m,REAL n,
    g be PartFunc of REAL-NS m,REAL-NS n,
    Z be set st
 f=g holds
  f is_partial_differentiable_on Z,i
     iff
  g is_partial_differentiable_on Z,i
proof
   let m,n be non zero Nat,
       i be Nat,
       f be PartFunc of REAL m,REAL n,
       g be PartFunc of REAL-NS m,REAL-NS n,
       Z be set;
   assume A1: f=g;
   hereby assume A2: f is_partial_differentiable_on Z,i;
then A3: Z c=dom f & for x be Element of REAL m st x in Z holds
     f|Z is_partial_differentiable_in x,i;
     now let y be Point of REAL-NS m;
     assume A4: y in Z;
     reconsider x=y as Element of REAL m by REAL_NS1:def 4;
      f|Z is_partial_differentiable_in x,i by A2,A4;
     then ex gZ be PartFunc of REAL-NS m,REAL-NS n, y be Point of REAL-NS m
      st f|Z=gZ & x=y & gZ is_partial_differentiable_in y,i;
     hence g|Z is_partial_differentiable_in y,i by A1;
    end;
    hence g is_partial_differentiable_on Z,i by A1,A3;
   end;
   assume A5: g is_partial_differentiable_on Z,i;
then A6:Z c=dom g & for y be Point of REAL-NS m st y in Z holds
     g|Z is_partial_differentiable_in y,i;
    now let x be Element of REAL m;
    assume A7:x in Z;
    reconsider y=x as Point of REAL-NS m by REAL_NS1:def 4;
     g|Z is_partial_differentiable_in y,i by A5,A7;
    hence f|Z is_partial_differentiable_in x,i by A1;
   end;
   hence f is_partial_differentiable_on Z,i by A1,A6;
end;
