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

theorem Th70:
for m be non zero Element of NAT, Z be set, i be Nat,
    f be PartFunc of REAL m,REAL holds
  f is_partial_differentiable_on Z,i iff f|Z is_partial_differentiable_on Z,i
proof
   let m be non zero Element of NAT, Z be set, i be Nat,
       f be PartFunc of REAL m,REAL;
   hereby assume A1: f is_partial_differentiable_on Z,i;
    (f|Z) |Z = f|Z by RELAT_1:72;
    hence f|Z is_partial_differentiable_on Z,i by A1,RELAT_1:62;
   end;
   assume A2: f|Z is_partial_differentiable_on Z,i;
   dom (f|Z) c= dom f by RELAT_1:60; then
A3:Z c=dom f by A2;
   (f|Z) |Z = f|Z by RELAT_1:72;
   hence f is_partial_differentiable_on Z,i by A2,A3;
end;
