reserve p1, p2 for Point of TOP-REAL 2,
  C for Simple_closed_curve,
  P for Subset of TOP-REAL 2;

theorem Th13:
  for T1,T2 being non empty TopSpace, f being Function of T1,T2 st
  f is continuous for A being Subset of T1, g being Function of T1|A, T2|(f.:A)
  st g = f|A holds g is continuous
proof
  let T1,T2 be non empty TopSpace;
  let f be Function of T1,T2;
  assume
A1: f is continuous;
  let A be Subset of T1;
  let g be Function of T1|A, T2|(f.:A);
  assume
A2: g = f|A;
A3: dom f = the carrier of T1 by FUNCT_2:def 1;
A4: [#](T1|A) = A by PRE_TOPC:def 5;
  per cases;
  suppose
    A is empty;
    hence thesis by TIETZE:4;
  end;
  suppose
    A is non empty;
    then reconsider S1 = T1|A, S2 = T2|(f.:A) as non empty TopSpace by A3;
    f|A = f|(T1|A) by A4,TMAP_1:def 3;
    then
A5: g is continuous Function of S1, T2 by A1,A2;
    g is Function of S1,S2;
    hence thesis by A5,JORDAN16:8;
  end;
end;
