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 Th8: :: CSSPACE2:5
  x^2+y^2>=2*|.x.|*|.y.|
proof
A1: x^2>=0 & y^2>=0 by XREAL_1:63;
  then x^2+y^2>=2*sqrt(x^2*y^2) by SIN_COS2:1;
  then x^2+y^2>=2*(sqrt(x^2)*sqrt(y^2)) by A1,SQUARE_1:29;
  then x^2+y^2>=2*(sqrt(|.x.|^2)*sqrt(y^2)) by COMPLEX1:75;
  then x^2+y^2>=2*(sqrt(|.x.|^2)*sqrt(|.y.|^2)) by COMPLEX1:75;
  then x^2+y^2>=2*(|.x.|*sqrt(|.y.|^2)) by SQUARE_1:22;
  then x^2+y^2>=2*(|.x.|*|.y.|) by SQUARE_1:22;
  hence thesis;
end;
