reserve a,b,c for positive Real,
  m,x,y,z for Real,
  n for Nat,
  s,s1,s2,s3,s4,s5 for Real_Sequence;

theorem
  a>=1 implies (a to_power b)+(a to_power c)>=2*(a to_power sqrt(b*c))
proof
A1: (b+c)/2>=(2*sqrt(b*c))/2 by SIN_COS2:1,XREAL_1:72;
  set p=a to_power c;
  set o=a to_power b;
  o>0 & p>0 by POWER:34;
  then
  (a to_power b)+(a to_power c)>=2*sqrt((a to_power b)*(a to_power c)) by
SIN_COS2:1;
  then
  (a to_power (b+c))>0 & (a to_power b)+(a to_power c)>=2*(sqrt(a to_power
  (b+ c))) by POWER:27,34;
  then
  (a to_power b)+(a to_power c)>=2*((a to_power (b+c)) to_power (1/2)) by
ASYMPT_1:83;
  then
A2: (a to_power b)+(a to_power c)>=2*((a to_power ((1/2)*(b+c)))) by POWER:33;
  assume a>=1;
  then a #R ((b+c)/2) >= a #R (sqrt(b*c)) by A1,PREPOWER:82;
  then a to_power ((b+c)/2) >= a #R (sqrt(b*c)) by POWER:def 2;
  then a to_power ((b+c)/2) >= a to_power (sqrt(b*c)) by POWER:def 2;
  then 2*(a to_power ((b+c)/2)) >= 2*(a to_power (sqrt(b*c))) by XREAL_1:64;
  hence thesis by A2,XXREAL_0:2;
end;
