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^2+a*b+b^2)/3)+sqrt((b^2+b*c+c^2)/3)+sqrt((c^2+c*a+a^2)/3) <=
  sqrt((a^2+b^2)/2)+sqrt((b^2+c^2)/2)+sqrt((c^2+a^2)/2)
proof
A1: sqrt((b^2+b*c+c^2)/3)<=sqrt((b^2+c^2)/2) by Lm12;
A2: sqrt((c^2+c*a+a^2)/3)<=sqrt((c^2+a^2)/2) by Lm12;
  sqrt((a^2+a*b+b^2)/3)<=sqrt((a^2+b^2)/2) by Lm12;
  then
  sqrt((a^2+a*b+b^2)/3)+sqrt((b^2+b*c+c^2)/3)<=sqrt((a^2+b^2)/2)+sqrt ((b
  ^2+c^2)/2) by A1,XREAL_1:7;
  hence thesis by A2,XREAL_1:7;
end;
