
theorem
  for T being SubSpace of TOP-REAL 2 st the carrier of T is
  Simple_closed_curve holds T is pathwise_connected
proof
  let T be SubSpace of TOP-REAL 2;
  assume
A1: the carrier of T is Simple_closed_curve;
  then reconsider P = the carrier of T as Simple_closed_curve;
  let a, b be Point of T;
  a in P & b in P;
  then reconsider p1 = a, p2 = b as Point of TOP-REAL 2;
  per cases;
  suppose
    p1 <> p2;
    then consider P1, P2 being non empty Subset of TOP-REAL 2 such that
A2: P1 is_an_arc_of p1,p2 and
    P2 is_an_arc_of p1,p2 and
A3: P = P1 \/ P2 and
    P1 /\ P2 = {p1,p2} by TOPREAL2:5;
    the carrier of ((TOP-REAL 2)|P1) = P1 by PRE_TOPC:8;
    then
A4: the carrier of ((TOP-REAL 2)|P1) c= P by A3,XBOOLE_1:7;
    then
A5: (TOP-REAL 2)|P1 is SubSpace of T by TSEP_1:4;
    consider f1 being Function of I[01], (TOP-REAL 2)|P1 such that
A6: f1 is being_homeomorphism and
A7: f1.0 = p1 & f1.1 = p2 by A2,TOPREAL1:def 1;
    reconsider f = f1 as Function of I[01],T by A4,FUNCT_2:7;
    take f;
    f1 is continuous by A6;
    hence f is continuous by A5,PRE_TOPC:26;
    thus thesis by A7;
  end;
  suppose
A8: p1 = p2;
    reconsider T1 = T as non empty SubSpace of TOP-REAL 2 by A1;
    reconsider a1 = a as Point of T1;
    reconsider f = I[01] --> a1 as Function of I[01],T;
    take f;
    thus f is continuous;
    thus f.0 = f.j0 .= a;
    thus f.1 = f.j1 .= b by A8;
  end;
end;
