reserve GX,GY for non empty TopSpace,
  x,y for Point of GX,
  r,s for Real;

theorem Th2:  ::Arcwise connectedness derives connectedness for subset
  for A being Subset of GX st
  (for xa,ya being Point of GX st xa in A & ya in A & xa<>ya
  ex h being Function of I[01],GX|A st h is continuous & xa=h.0 & ya=h.1)
  holds A is connected
proof
  let A be Subset of GX;
  assume that
A1: for xa,ya being Point of GX st xa in A & ya in A & xa<>ya
  ex h being Function of I[01],GX|A st h is continuous & xa=h.0 & ya=h.1;
  per cases;
  suppose A is non empty;
    then reconsider A as non empty Subset of GX;
A2: for xa,ya being Point of GX st xa in A & ya in A & xa = ya
    ex h being Function of I[01],GX|A st h is continuous & xa=h.0 & ya=h.1
    proof
      let xa,ya be Point of GX;
      assume that
A3:   xa in A and ya in A and
A4:   xa = ya;
      reconsider xa9 = xa as Element of GX|A by A3,PRE_TOPC:8;
      reconsider h = I[01] --> xa9 as Function of I[01], GX|A;
      take h;
      thus thesis by A4,Lm1,BORSUK_1:40,FUNCOP_1:7;
    end;
    for xb,yb being Point of GX|A
    ex ha being Function of I[01],GX|A st ha is continuous & xb=ha.0 & yb=ha.1
    proof
      let xb,yb be Point of GX|A;
A5:   xb in [#](GX|A);
A6:   yb in [#](GX|A);
A7:   xb in A by A5,PRE_TOPC:def 5;
A8:   yb in A by A6,PRE_TOPC:def 5;
      per cases;
      suppose xb<>yb;
        hence thesis by A1,A7,A8;
      end;
      suppose xb = yb;
        hence thesis by A2,A7;
      end;
    end;
    then GX|A is connected by Th1;
    hence thesis;
  end;
  suppose A is empty;
    then reconsider D = A as empty Subset of GX;
    let A1, B1 be Subset of GX|A such that
    [#](GX|A) = A1 \/ B1 and A1,B1 are_separated;
    [#](GX|D) = D;
    hence thesis;
  end;
end;
