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

theorem Th19:
  cosh.(-p) = cosh.p & sinh.(-p) = -sinh.p & tanh.(-p) = -tanh.p
proof
A1: cosh.(-p) = (exp_R.(-p) + exp_R.(-(-p)))/2 by Def3
    .= cosh.p by Def3;
A2: sinh.(-p) = (exp_R.(-p) - exp_R.(-(-p)))/2 by Def1
    .= -(exp_R.(p) - exp_R.(-p))/2
    .= -sinh.p by Def1;
  then tanh.(-p) = (-sinh.p)/(cosh.(-p)) by Th17
    .= -(sinh.p)/(cosh.p) by A1,XCMPLX_1:187
    .= -tanh.p by Th17;
  hence thesis by A1,A2;
end;
