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

theorem Th61:
for X be Subset of REAL m,
    f be PartFunc of REAL m,REAL, g be PartFunc of REAL m,REAL 1, i being Nat
  st
 <>*f = g & X is open & 1 <= i & i <= m holds
  f is_partial_differentiable_on X,i iff g is_partial_differentiable_on X,i
proof
   let X be Subset of REAL m;
   let f be PartFunc of REAL m,REAL, g be PartFunc of REAL m,REAL 1,
       i be Nat;
   assume A1: <>*f = g & X is open & 1 <= i & i <= m;
   hereby assume A2: f is_partial_differentiable_on X,i; then
A3: X c= dom g by Th3,A1;
    now let x be Element of REAL m;
     assume x in X; then
     f is_partial_differentiable_in x,i by A2,A1,Th60;
     hence g is_partial_differentiable_in x,i by A1,PDIFF_1:18;
    end;
    hence g is_partial_differentiable_on X,i by A3,A1,PDIFF_7:34;
   end;
   hereby assume
A4: g is_partial_differentiable_on X,i; then
A5: X c= dom f by Th3,A1;
    now let x be Element of REAL m;
     assume x in X; then
     g is_partial_differentiable_in x,i by A4,A1,PDIFF_7:34;
     hence f is_partial_differentiable_in x,i by A1,PDIFF_1:18;
    end;
    hence f is_partial_differentiable_on X,i by A1,Th60,A5;
   end;
end;
