reserve p,q,r,th,th1 for Real;
reserve n for Nat;

theorem
  tanh.p<1 & tanh.p>-1
proof
A1: exp_R.p > 0 & exp_R.(-p) > 0 by SIN_COS:54;
  thus tanh.p<1
  proof
    assume tanh.p>=1;
    then
A2: (exp_R.p - exp_R.(-p))/(exp_R.p + exp_R.(-p))>=1 by Def5;
    exp_R.p > 0 & exp_R.(-p) > 0 by SIN_COS:54;
    then
A3: (exp_R.p - exp_R.(-p))/(exp_R.p + exp_R.(-p))*(exp_R.p + exp_R.(-p)) =
    exp_R.p - exp_R.(-p) by XCMPLX_1:87;
    (exp_R.p + exp_R.-p) >= 2 by Lm24;
    then
    (exp_R.p - exp_R.(-p))/(exp_R.p + exp_R.(-p))*(exp_R.p + exp_R.(-p ))
    >=1*(exp_R.p + exp_R.(-p)) by A2,XREAL_1:64;
    then (exp_R.p - exp_R.(-p))-exp_R.p >= (exp_R.p + exp_R.(-p))-exp_R.p by A3
,XREAL_1:9;
    then
A4: -exp_R.(-p)+exp_R.(-p) >= exp_R.(-p)+exp_R.(-p) by XREAL_1:6;
    exp_R.(-p) > 0 by SIN_COS:54;
    hence contradiction by A4;
  end;
  assume tanh.p<=-1;
  then
A5: (exp_R.p - exp_R.(-p))/(exp_R.p + exp_R.(-p))<=-1 by Def5;
  (exp_R.p + exp_R.-p) >= 2 by Lm24;
  then
  (exp_R.p - exp_R.(-p))/(exp_R.p + exp_R.(-p))*(exp_R.p + exp_R.(-p)) <=
  (-1)*(exp_R.p + exp_R.(-p)) by A5,XREAL_1:64;
  then exp_R.p - exp_R.(-p)<=-(exp_R.p + exp_R.(-p)) by A1,XCMPLX_1:87;
  then
A6: exp_R.p - exp_R.(-p)+exp_R.(-p) <=-exp_R.p + -exp_R.(-p)+exp_R.(-p) by
XREAL_1:6;
  exp_R.p > 0 by SIN_COS:54;
  hence contradiction by A6;
end;
