reserve x,x1,x2,x3 for Real;

theorem
  x<=0 implies sinh x<=0
proof
  assume
A1: x<=0;
  per cases by A1,XXREAL_0:1;
  suppose
A2: x < 0;
    then -x>0 by XREAL_1:58;
    then
A3: exp_R.(-x)>=1 by SIN_COS:52;
    exp_R.x<=1 by A2,SIN_COS:53;
    then exp_R.x-exp_R.(-x)<=1-1 by A3,XREAL_1:13;
    then (exp_R.x-exp_R.(-x))/2<=0 by XREAL_1:138;
    then sinh.x<=0 by SIN_COS2:def 1;
    hence thesis by SIN_COS2:def 2;
  end;
  suppose
    x = 0;
    hence thesis by SIN_COS2:16,def 2;
  end;
end;
