
theorem NEW:
  for a,b,c,n be positive Real, m be non negative Real holds
  a to_power m + b to_power m <= c to_power m implies
  a to_power (m+n) + b to_power (m+n) < c to_power (m+n)
  proof
    let a,b,c,n be positive Real, m be non negative Real;
    assume
    A1: a to_power m + b to_power m <= c to_power m;
    reconsider x = a/c as positive Real;
    reconsider y = b/c as positive Real;
    A2: x*c = a & y*c = b by XCMPLX_1:87;
    A3: (x*c) to_power m = (x to_power m)*(c to_power m) &
    (y*c) to_power m = (y to_power m)*(c to_power m) &
    (x*c) to_power (m+n) = (x to_power (m+n))*(c to_power (m+n)) &
    (y*c) to_power (m+n) = (y to_power (m+n))*(c to_power (m+n))
      by POWER:30; then
    (c to_power m)*(x to_power m + y to_power m) <= (c to_power m)*1
      by A1,A2; then
    x to_power m + y to_power m <= 1 by XREAL_1:68; then
    x to_power (m+n) + y to_power (m+n) < 1 by Pow; then
    (c to_power (m+n))*(x to_power (m+n) + y to_power (m+n)) <
      (c to_power (m+n))*1 by XREAL_1:68;
    hence thesis by A2,A3;
  end;
