reserve a, b, c, d, r, s for Real,
  n for Element of NAT,
  p, p1, p2 for Point of TOP-REAL 2,
  x, y for Point of TOP-REAL n,
  C for Simple_closed_curve,
  A, B, P for Subset of TOP-REAL 2,
  U, V for Subset of (TOP-REAL 2)|C`,
  D for compact with_the_max_arc Subset of TOP-REAL 2;

theorem Th1:
  for p1, p2 being Point of TOP-REAL n st p1 <> p2 holds 1/2*(p1+p2) <> p1
proof
  let p1, p2 be Point of TOP-REAL n;
  set r = 1/2;
  assume that
A1: p1 <> p2 and
A2: r*(p1+p2) = p1;
  r*(p1+p2) = r*p1+r*p2 by RLVECT_1:def 5;
  then 0.TOP-REAL n = p1-(r*p1+r*p2) by A2,RLVECT_1:5
    .= p1-r*p1-r*p2 by RLVECT_1:27
    .= 1 * p1-r*p1-r*p2 by RLVECT_1:def 8
    .= (1-r)*p1-r*p2 by RLVECT_1:35
    .= r*(p1-p2) by RLVECT_1:34;
  then p1-p2 = 0.TOP-REAL n by RLVECT_1:11;
  hence thesis by A1,RLVECT_1:21;
end;
