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+y+z=1 implies x*y+y*z+x*z<=1/3
proof
  x^2+y^2>=2*x*y & x^2+z^2>=2*x*z by Th6;
  then
A1: (x^2+y^2)+(x^2+z^2)>=2*x*y+2*x*z by XREAL_1:7;
  y^2+z^2>=2*y*z by Th6;
  then (x^2+y^2+x^2+z^2)+(y^2+z^2)>=2*x*y+2*x*z+2*y*z by A1,XREAL_1:7;
  then 2*(x^2+y^2+z^2)/2>=2*(x*y+x*z+y*z)/2 by XREAL_1:72;
  then
A2: x^2+y^2+z^2+2*(x*z+y*z+x*y)>=x*y+x*z+y*z+2*(x*z+y*z+x*y)by XREAL_1:7;
  assume x+y+z =1;
  then 1^2=(x+y)^2+2*(x+y)*z+z^2 by SQUARE_1:4
    .=x^2+y^2+z^2+2*(x*z+y*z+x*y);
  then 1/3>=3*(x*z+y*z+x*y)/3 by A2,XREAL_1:72;
  hence thesis;
end;
