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
  x+y+z=1 implies x*y+y*z+z*x<=1/3
proof
  x^2+y^2+z^2>=x*y+y*z+x*z by SERIES_3:10;
  then
A1: (x^2+y^2+z^2)+2*(x*y+y*z+z*x)>=(x*y+y*z+x*z)+2*(x*y+y*z+z*x) by XREAL_1:7;
  assume x+y+z=1;
  then 1^2=x^2+y^2+z^2+2*x*y+2*y*z+2*z*x by Lm6;
  then 1/3>=(3*(x*y+y*z+z*x))/3 by A1,XREAL_1:72;
  hence thesis;
end;
