
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 iff a = b
  proof
    let a,b be positive Real, n,m be positive Real;
    a = b implies (a to_power n - b to_power n)*(a to_power m - b to_power m)
      = 0; then
    A1: a = b implies 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) + 0)/2
        by N158;
    a <> b implies
      (a to_power n - b to_power n)*(a to_power m - b to_power m) > 0
    proof
      assume
      a <> b; then
      per cases by XXREAL_0:1;
      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 <> b implies (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 <> b implies ((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_1:68;
    hence thesis by A1,N158;
  end;
