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+b^2)/2)+sqrt((b^2+c^2)/2)+sqrt((c^2+a^2)/2) <=sqrt(3*(a^2+b
  ^2+c^2))
proof
A1: (sqrt((a^2+b^2)/2))^2=(a^2+b^2)/2 by SQUARE_1:def 2;
A2: (sqrt((b^2+c^2)/2))^2=(b^2+c^2)/ 2 by SQUARE_1:def 2;
A3: sqrt((c^2+a^2)/2)>0 by SQUARE_1:25;
A4: sqrt((b^2+c^2)/2)>0 by SQUARE_1:25;
A5: (sqrt((a^2+b^2)/2)+sqrt((b^2+c^2)/2)+sqrt((c^2+a^2)/2))^2>=0 by XREAL_1:63;
A6: sqrt((a^2+b^2)/2)>0 by SQUARE_1:25;
A7: (sqrt((c^2+a^2)/2))^2=(c^2+a^2)/2 by SQUARE_1:def 2;
  (1*sqrt((a^2+b^2)/2)+1*sqrt((b^2+c^2)/2)+1*sqrt((c^2+a^2)/2))^2 <=(1^2+1
^2+1^2)*((sqrt((a^2+b^2)/2))^2+(sqrt((b^2+c^2)/2))^2+ (sqrt((c^2+a^2)/2))^2)
by Th29;
  then
  sqrt((sqrt((a^2+b^2)/2)+sqrt((b^2+c^2)/2)+sqrt((c^2+a^2)/2))^2)<= sqrt(
  3*(a^2+b^2+c^2)) by A1,A2,A7,A5,SQUARE_1:26;
  hence thesis by A6,A4,A3,SQUARE_1:22;
end;
