reserve a,b,c,d for positive Real,
  m,u,w,x,y,z for Real,
  n,k for Nat,
  s,s1 for Real_Sequence;

theorem
  sqrt(a*b)<=sqrt((a^2+b^2)/2)
proof
A1: (a+b)/2>=sqrt(a*b) by SERIES_3:2;
  (a+b)/2<=sqrt((a^2+b^2)/2) by Th8;
  hence thesis by A1,XXREAL_0:2;
end;
