
theorem ::see NEWTON03:42;
  for a,b,n be non negative Real holds
  max (a to_power n, b to_power n) = max (a,b) to_power n &
  min (a to_power n, b to_power n) = min (a,b) to_power n
  proof
    let a,b,n be non negative Real;
    per cases;
    suppose n is zero;
      then max (a,b) to_power n = 1 & min (a,b) to_power n = 1 &
        a to_power n = 1 & b to_power n = 1 by POWER:24;
      hence thesis;
    end;
    suppose n is non zero; then
      reconsider n as positive Real;
      per cases;
      suppose
        B1: a >= b; then
        B2: max (a,b) = a & min (a,b) = b by XXREAL_0:def 9, XXREAL_0:def 10;
        a to_power n >= b to_power n by B1,POWER37;
        hence thesis by B2,XXREAL_0:def 9,XXREAL_0:def 10;
      end;
      suppose
        B1: a < b; then
        B2: max (a,b) = b & min (a,b) = a by XXREAL_0:def 9, XXREAL_0:def 10;
        a to_power n < b to_power n by B1,POWER37;
        hence thesis by B2,XXREAL_0:def 9,XXREAL_0:def 10;
      end;
    end;
  end;
