
theorem :: Problem 154
  not ex x,y,z being positive Nat st
    x / y + y / z + z / x = 2
  proof
    given x,y,z being positive Nat such that
A1: x / y + y / z + z / x = 2;
    per cases;
    suppose
A2:   x / y = y / z & y / z = z / x;
      (x / y) * (y / z) * (z / x) = 1 by Multiply1; then
      (x / y) |^ 2 * (z / x) = 1 by A2,NEWTON:81; then
      (x / y) |^ (2+1) = 1 by A2,NEWTON:6; then
      x / y = 3-Root 1 by PREPOWER:19; then
      x / y = 1 by PREPOWER:20;
      hence thesis by A1,A2;
    end;
    suppose
D1:   x / y <> y / z or y / z <> z / x;
      set a = x / y, b = y / z, c = z / x;
      ((a + b + c) / 3) |^ 3 > a * b * c by SERIES3,D1; then
      ((a + b + c) / 3) |^ 3 > 1 by Multiply1; then
      3-Root (((a + b + c) / 3) |^ 3) > 3-Root 1 by PREPOWER:28; then
      3-Root (((a + b + c) / 3) |^ 3) > 1 by PREPOWER:20; then
      (a + b + c) / 3 > 1 by PREPOWER:19;
      hence thesis by A1;
    end;
  end;
