theorem Th19:
  for P being Subset of TOP-REAL n st P=REAL n holds P is connected
proof
  let P be Subset of TOP-REAL n;
  assume
A1: P=(REAL n);
  for p1,p2 being Point of TOP-REAL n st p1 in P & p2 in P holds LSeg(p1,
  p2) c= P
  proof
    let p1,p2 be Point of TOP-REAL n;
    assume that
    p1 in P and
    p2 in P;
    the carrier of TOP-REAL n=REAL n by EUCLID:22;
    hence thesis by A1;
  end;
  then P is convex by JORDAN1:def 1;
  hence thesis;
