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=1 implies x^2+y^2>=1/2
proof
  (x-y)^2 >= 0 by XREAL_1:63;
  then
A1: x^2-2*x*y+y^2+(x^2+2*x*y+y^2) >= 0+(x^2+2*x*y+y^2)by XREAL_1:7;
  assume x+y=1;
  then 1^2=x^2+2*x*y+y^2 by SQUARE_1:4;
  then 2*(x^2+y^2)/2 >= 1/2 by A1,XREAL_1:72;
  hence thesis;
end;
