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
  a+b+c<=sqrt((a^2+a*b+b^2)/3)+sqrt((b^2+b*c+c^2)/3)+sqrt((c^2+c*a+a^2)/ 3)
proof
A1: sqrt(3)>0 by SQUARE_1:25;
A2: sqrt(c^2+c*a+a^2)>=(1/2)*sqrt(3)*(c+a) by Lm11;
A3: sqrt(b^2+b*c+c^2)>=(1/2)*sqrt(3)*(b+c) by Lm11;
  sqrt(a^2+a*b+b^2)>=(1/2)*sqrt(3)*(a+b) by Lm11;
  then
  sqrt(a^2+a*b+b^2)+sqrt(b^2+b*c+c^2)>=(1/2)*sqrt(3)*(a+b)+(1/2)*sqrt( 3)*
  (b+c) by A3,XREAL_1:7;
  then
  sqrt(a^2+a*b+b^2)+sqrt(b^2+b*c+c^2)+sqrt(c^2+c*a+a^2)>=(1/2)*sqrt(3)*( a
  +b)+ (1/2)*sqrt(3)*(b+c)+(1/2)*sqrt(3)*(c+a) by A2,XREAL_1:7;
  then
  (sqrt(a^2+a*b+b^2)+sqrt(b^2+b*c+c^2)+sqrt(c^2+c*a+a^2))/sqrt(3)>=((a+b +
  c)* sqrt(3))/sqrt(3) by A1,XREAL_1:72;
  then
  (sqrt(a^2+a*b+b^2)+sqrt(b^2+b*c+c^2)+sqrt(c^2+c*a+a^2))/sqrt(3)>=(a+b+ c
  )* (sqrt(3)/sqrt(3)) by XCMPLX_1:74;
  then
  (sqrt(a^2+a*b+b^2)+sqrt(b^2+b*c+c^2)+sqrt(c^2+c*a+a^2))/sqrt(3)>=(a+b+ c
  )*1 by A1,XCMPLX_1:60;
  then (sqrt(a^2+a*b+b^2)+sqrt(b^2+b*c+c^2))/sqrt(3)+sqrt(c^2+c*a+a^2)/sqrt(3
  ) >=a+b+c by XCMPLX_1:62;
  then
  sqrt(a^2+a*b+b^2)/sqrt(3)+sqrt(b^2+b*c+c^2)/sqrt(3)+sqrt(c^2+c*a+a^2 )/
  sqrt(3)>=a+b+c by XCMPLX_1:62;
  then
  sqrt((a^2+a*b+b^2)/3)+sqrt(b^2+b*c+c^2)/sqrt(3)+sqrt(c^2+c*a+a^2)/ sqrt
  (3)>=a+b+c by SQUARE_1:30;
  then sqrt((a^2+a*b+b^2)/3)+sqrt((b^2+b*c+c^2)/3)+sqrt(c^2+c*a+a^2)/ sqrt(3)
  >=a+b+c by SQUARE_1:30;
  hence thesis by SQUARE_1:30;
end;
