reserve x,y,t for Real;

theorem
  x^2<1 implies ((2*x)/(1+x^2))^2<1
proof
  assume x^2<1;
  then x^2+(-1)<1+(-1) by XREAL_1:8;
  then (x^2-1)*(x^2-1)>0*(x^2-1);
  then
A1: x^2>=0 & (x^2)^2-2*(x^2)+1+4*x^2>0+4*x^2 by XREAL_1:8,63;
  ((2*x)/(1+x^2))^2 = (2*x)^2/(1+x^2)^2 by XCMPLX_1:76
    .=4*x^2/((x^2)^2+2*(x^2)+1);
  hence thesis by A1,XREAL_1:189;
end;
