theorem Th4:
  ((x+y)/2)^2>=x*y
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+2*x*y>=2*x*y+2*x*y by XREAL_1:7;
  then (x+y)^2/(2*2)>=4*(x*y)/4 by XREAL_1:72;
  hence thesis;
end;
