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;
