
theorem Pow:
  for a,b be positive Real, m be non negative Real, n be positive Real holds
  a to_power m + b to_power m <= 1 implies
  a to_power (m+n) + b to_power (m+n) < 1
  proof
    let a,b be positive Real, m be non negative Real,n be positive Real;
    assume
    A1: a to_power m + b to_power m <= 1;
    a to_power 0 = 1 & b to_power 0 = 1 by POWER:24; then
    not m = 0 by A1; then
    reconsider m as positive Real;
    A2: a to_power m + 0 < a to_power m + b to_power m by XREAL_1:6;
    (a > 1 implies a to_power m >= 1) & (a = 1 implies a to_power m = 1)
      by POWER:35; then
    a >= 1 implies a to_power m >= 1 by XXREAL_0:1; then
    reconsider a as light positive Real by A1,A2,TA1,XXREAL_0:2;
    A3: b to_power m + 0  < b to_power m + a to_power m by XREAL_1:6;
    (b > 1 implies b to_power m >= 1) & (b = 1 implies b to_power m = 1)
      by POWER:35; then
    b >= 1 implies b to_power m >= 1 by XXREAL_0:1; then
    reconsider b as light positive Real by A1,A3,TA1,XXREAL_0:2;
    0 < a < 1 & 0 < b < 1 & m + 0 < m + n  by TA1,XREAL_1:6; then
    a to_power m > a to_power (m+n) & b to_power m > b to_power (m+n)
      by POWER:40; then
    a to_power (m+n) + b to_power (m+n) < a to_power m + b to_power m
      by XREAL_1:8;
    hence thesis by A1,XXREAL_0:2;
  end;
