reserve x,y,t for Real;

theorem Th5:
  sqrt(x^2+1)+x>0
proof
  per cases;
  suppose
    x>0;
    hence thesis by Th4,XREAL_1:29;
  end;
  suppose
A1: x<=0;
    x^2<x^2+1 by XREAL_1:29;
    then sqrt(x^2)<sqrt(x^2+1) by SQUARE_1:27,XREAL_1:63;
    then -x<sqrt(x^2+1) by A1,SQUARE_1:23;
    hence thesis by XREAL_1:62;
  end;
end;
