
theorem
  for a,b be positive Real, n,m be positive Real holds
  a to_power (n+m) + b to_power (n+m) >=
  (a to_power n + b to_power n)*(a to_power m + b to_power m)/2
  proof
    let a,b be positive Real, n,m be positive Real;
    (a to_power n - b to_power n)*(a to_power m - b to_power m) >= 0
    proof
      per cases by XXREAL_0:1;
      suppose
        a = b;
        hence thesis;
      end;
      suppose
        a > b; then
        a to_power n > b to_power n & a to_power m > b to_power m
          by POWER:37; then
        a to_power n - b to_power n > b to_power n - b to_power n &
        a to_power m - b to_power m > b to_power m - b to_power m
          by XREAL_1:9;
        hence thesis;
      end;
      suppose
        a < b; then
        a to_power n < b to_power n & a to_power m < b to_power m
          by POWER:37; then
        a to_power n - b to_power n < b to_power n - b to_power n &
          a to_power m - b to_power m < b to_power m - b to_power m
            by XREAL_1:9;
        hence thesis;
      end;
    end; then
    (a to_power m + b to_power m)*(a to_power n + b to_power n) +
      (a to_power n - b to_power n)*(a to_power m - b to_power m)
    >= (a to_power m + b to_power m)*(a to_power n + b to_power n) + 0
      by XREAL_1:6; then
    ((a to_power m + b to_power m)*(a to_power n + b to_power n) +
      (a to_power n - b to_power n)*(a to_power m - b to_power m))/2
    >= ((a to_power m + b to_power m)*(a to_power n + b to_power n))/2
      by XREAL_1:72;
    hence thesis by N158;
  end;
