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^2+y^2+z^2>=1/3
proof
A1: (y^2+z^2)/2>=y*z by SERIES_3:7;
A2: (x^2+z^2)/2>=x*z by SERIES_3:7;
  (x^2+y^2)/2>=x*y by SERIES_3:7;
  then (x^2+y^2)/2+(y^2+z^2)/2>=y*z+x*y by A1,XREAL_1:7;
  then (x^2+y^2)/2+(y^2+z^2)/2+(x^2+z^2)/2>=y*z+x*y+x*z by A2,XREAL_1:7;
  then -((x^2+y^2)/2+(y^2+z^2)/2+(x^2+z^2)/2)<=-(y*z+x*y+x*z) by XREAL_1:24;
  then (-((x^2+y^2)/2+(y^2+z^2)/2+(x^2+z^2)/2))*2<=(-(y*z+x*y+x*z))*2 by
XREAL_1:64;
  then -((x^2+y^2)/2+(y^2+z^2)/2+(x^2+z^2)/2)*2+1<=-(y*z+x*y+x*z)*2+1 by
XREAL_1:6;
  then
A3: 1-(x^2+y^2+z^2)*2<=1-(y*z+x*y+x*z)*2;
  assume x+y+z=1;
  then x^2+y^2+z^2+2*x*y+2*y*z+2*z*x=1^2 by Lm6;
  then 1<=(x^2+y^2+z^2)+(x^2+y^2+z^2)*2 by A3,XREAL_1:20;
  then 1/3<=((x^2+y^2+z^2)*3)/3 by XREAL_1:72;
  hence thesis;
end;
