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 Th8:
  (a+b)/2<=sqrt((a^2+b^2)/2)
proof
  a^2+b^2>=2*a*b by SERIES_3:6;
  then (a^2+b^2)+(a^2+b^2)>=2*a*b+(a^2+b^2) by XREAL_1:7;
  then sqrt(2*(a^2+b^2))>=sqrt((a+b)^2) by SQUARE_1:26;
  then sqrt(2*(a^2+b^2))>=(a+b) by SQUARE_1:22;
  then (sqrt(2*(a^2+b^2)))/sqrt(2*2)>=(a+b)/2 by SQUARE_1:20,XREAL_1:72;
  then (sqrt(2*(a^2+b^2)))/(sqrt(2)*sqrt(2))>=(a+b)/2 by SQUARE_1:29;
  then (sqrt(2)*sqrt(a^2+b^2))/(sqrt(2)*sqrt(2))>=(a+b)/2 by SQUARE_1:29;
  then
A1: (sqrt(2)/sqrt(2)*sqrt(a^2+b^2))/sqrt(2)>=(a+b)/2 by XCMPLX_1:83;
  sqrt(2)>0 by SQUARE_1:25;
  then (1*sqrt(a^2+b^2))/sqrt(2)>=(a+b)/2 by A1,XCMPLX_1:60;
  hence thesis by SQUARE_1:30;
end;
