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 Th10:
  x^2+y^2+z^2>=x*y+y*z+x*z
proof
  x^2+y^2>=2*x*y & y^2+z^2>=2*y*z by Th6;
  then
A1: x^2+y^2+(y^2+z^2)>=2*x*y+2*y*z by XREAL_1:7;
  z^2+x^2>=2*z*x by Th6;
  then (x^2+y^2+(y^2+z^2))+(z^2+x^2)>=(2*x*y+2*y*z)+(2*z*x) by A1,XREAL_1:7;
  then 2*(x^2+y^2+z^2)/2>=2*(x*y+y*z+z*x)/2 by XREAL_1:72;
  hence thesis;
end;
