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 Th34:
for m,n be non zero Nat, i be Nat,
     f be PartFunc of REAL m,REAL n,
     Z be Subset of REAL m
     st Z is open & 1 <= i & i <= m holds
   f is_partial_differentiable_on Z,i
iff
   Z c=dom f &
   for x be Element of REAL m st x in Z holds
     f is_partial_differentiable_in x,i
proof
   let m,n be non zero Nat,
       i be Nat,
       f be PartFunc of REAL m,REAL n,
       Z be Subset of REAL m;
   assume A1: Z is open & 1 <= i & i <= m;
   then consider Z0 be Subset of REAL-NS m such that
A2: Z=Z0 & Z0 is open;
    the carrier of REAL-NS m = REAL m &
   the carrier of REAL-NS n = REAL n by REAL_NS1:def 4;
   then reconsider g=f as PartFunc of REAL-NS m,REAL-NS n;
   hereby assume f is_partial_differentiable_on Z,i;
then A3: g is_partial_differentiable_on Z0,i by A2,Th33;
    hence Z c=dom f by A2;
    thus for x be Element of REAL m st x in Z
           holds f is_partial_differentiable_in x,i
    proof
     let x be Element of REAL m;
     assume A4: x in Z;
     reconsider y=x as Point of REAL-NS m by REAL_NS1:def 4;
      g is_partial_differentiable_in y,i by A2,A3,A4,A1,Th8;
     hence f is_partial_differentiable_in x,i;
    end;
   end;
   assume A5: Z c=dom f & for x be Element of REAL m st x in Z
                 holds f is_partial_differentiable_in x,i;
    now let y be Point of REAL-NS m;
    assume A6: y in Z0;
    reconsider x=y as Element of REAL m by REAL_NS1:def 4;
     f is_partial_differentiable_in x,i by A2,A6,A5;
    then ex gZ be PartFunc of REAL-NS m,REAL-NS n, y be Point of REAL-NS m
     st f=gZ & x=y & gZ is_partial_differentiable_in y,i;
    hence g is_partial_differentiable_in y,i;
   end;
   then g is_partial_differentiable_on Z0,i by A1,Th8,A2,A5;
   hence f is_partial_differentiable_on Z,i by Th33,A2;
end;
