
theorem SERIES3: :: see SERIES_3:16
  for a,b,c being positive Real st
    not (a = b & b = c) holds
      ((a + b + c) / 3) |^ 3 > a * b * c
  proof
    let a,b,c be positive Real;
    assume
H1: not (a = b & b = c);
    set u = 3-root a;
    set v = 3-root b;
    set w = 3-root c;
    reconsider u,v,w as positive Real by PositiveRoot;
A1: a = u |^ 3 & b = v |^ 3 & c = w |^ 3;
T1: (u - v) |^ 2 = (u - v) ^2 & (v - w) |^ 2 = (v - w) ^2 &
      (w - u) |^ 2 = (w - u) ^2 by NEWTON:81;
T3: (u - v) |^ 2 + (v - w) |^ 2 + (w - u) |^ 2 > 0
    proof
      assume (u - v) |^ 2 + (v - w) |^ 2 + (w - u) |^ 2 <= 0; then
      (u - v) = 0 & (v - w) = 0 & (w - u) = 0;
      hence thesis by A1,H1;
    end;
    (u + v + w) / 2 * ((u - v) |^ 2 + (v - w) |^ 2 + (w - u) |^ 2) =
         (u ^2 * u + v ^2 * u + w ^2 * u - u * (u * v + w * u + v * w)) +
         (u ^2 * v + v ^2 * v + w ^2 * v - v * (u * v + w * u + v * w)) +
         (u ^2 * w + v ^2 * w + w ^2 * w - w * (u * v + w * u + v * w)) by T1
      .= (u ^2 * u + v ^2 * u + w ^2 * u - u * (u * v + w * u + v * w)) +
         (u ^2 * v + v |^ 3 + w ^2 * v - v * (u * v + w * u + v * w)) +
         (u ^2 * w + v ^2 * w + w ^2 * w - w * (u * v + w * u + v * w))
             by POLYEQ_2:4
      .= (u |^ 3 + v ^2 * u + w ^2 * u - u * (u * v + w * u + v * w)) +
         (u ^2 * v + v |^ 3 + w ^2 * v - v * (u * v + w * u + v * w)) +
         (u ^2 * w + v ^2 * w + w ^2 * w - w * (u * v + w * u + v * w))
             by POLYEQ_2:4
      .= (u |^ 3 + v ^2 * u + w ^2 * u - u * (u * v + w * u + v * w)) +
         (u ^2 * v + v |^ 3 + w ^2 * v - v * (u * v + w * u + v * w)) +
         (u ^2 * w + v ^2 * w + w |^ 3 - w * (u * v + w * u + v * w))
             by POLYEQ_2:4
      .= u |^ 3 + v |^ 3 + w |^ 3 - 3 * u * v * w; then
    u |^ 3 + v |^ 3 + w |^ 3 - 3 * u * v * w + 3 * u * v * w >
       0 + 3 * u * v * w by T3,XREAL_1:8; then
    (u |^ 3 + v |^ 3 + w |^ 3) / 3 > (3 * u * v * w) / 3 by XREAL_1:74; then
    ((u |^ 3 + v |^ 3 + w |^ 3) / 3) |^ 3 > (u * v * w) |^ 3
       by PREPOWER:10; then
    ((u |^ 3 + v |^ 3 + w |^ 3) / 3) |^ 3 > (u * v) |^ 3 * (w |^ 3)
       by NEWTON:7;
    hence thesis by NEWTON:7;
  end;
