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 Th3:
  p1`2 < p2`2 implies (1/2*(p1+p2))`2 < p2`2
proof
  assume
A1: p1`2 < p2`2;
  (1/2*(p1+p2))`2 = 1/2*((p1+p2)`2) by TOPREAL3:4
    .= 1/2*(p1`2+p2`2) by TOPREAL3:2
    .= (p1`2+p2`2)/2;
  hence thesis by A1,XREAL_1:226;
end;
