reserve a,b,c for positive Real,
  m,x,y,z for Real,
  n for Nat,
  s,s1,s2,s3,s4,s5 for Real_Sequence;

theorem
  (x^2+y^2)/2>=((x+y)/2)^2
proof
  (x-y)^2>=0 by XREAL_1:63;
  then x^2-2*x*y+y^2+2*x*y>=0+2*x*y by XREAL_1:7;
  then x^2+y^2+(x^2+y^2)>=2*x*y+(x^2+y^2) by XREAL_1:7;
  then 2*(x^2+y^2)/4>=(x+y)^2/4 by XREAL_1:72;
  hence thesis;
end;
