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

theorem Th31:
  sinh is_differentiable_in p & diff(sinh,p)=cosh.p
proof
  set ff = compreal;
A1: sinh = (1/2)(#)(exp_R-exp_R*ff) by Lm18;
  diff(sinh,p) =diff( ((1/2)(#)(exp_R-exp_R*ff)),p ) by Lm18
    .=(1/2)*diff((exp_R-(exp_R*ff)),p) by Lm16
    .=(1/2)*( exp_R.p+exp_R.(-p) ) by Lm15
    .=( exp_R.p+exp_R.(-p) )/2
    .=cosh.p by Def3;
  hence thesis by A1,Lm16;
end;
