reserve x,y,t for Real;

theorem
  (x/sqrt(x^2+1))^2<1
proof
A1: x^2<x^2+1 by XREAL_1:29;
A2: x^2>=0 by XREAL_1:63;
  then (sqrt(x^2+1))^2=sqrt(x^2+1)^2 by SQUARE_1:29
    .=x^2+1 by A2,SQUARE_1:22;
  then
A3: (x/sqrt(x^2+1))^2=(x^2)/(x^2+1) by XCMPLX_1:76;
  per cases by XREAL_1:63;
  suppose
    x^2>0;
    hence thesis by A1,A3,XREAL_1:189;
  end;
  suppose
    x^2=0;
    hence thesis;
  end;
end;
