reserve m,n,i,i2,j for Nat,
  r,r1,r2,s,t for Real,
  x,y,z for object;
reserve p,p1,p2,p3,q,q1,q2,q3,q4 for Point of TOP-REAL n;
reserve u for Point of Euclid n;

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;
