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

theorem Th32:
  cosh is_differentiable_in p & diff(cosh,p)=sinh.p
proof
  reconsider ff = compreal as PartFunc of REAL,REAL;
A1: cosh = (1/2)(#)(exp_R+exp_R*ff) by Lm21;
  diff(cosh,p) =diff( ((1/2)(#)(exp_R+exp_R*ff)),p ) by Lm21
    .=(1/2)*diff((exp_R+(exp_R*ff)),p) by Lm19
    .=(1/2)*( exp_R.p-exp_R.(-p) ) by Lm15
    .=( exp_R.p-exp_R.(-p) )/2
    .=sinh.p by Def1;
  hence thesis by A1,Lm19;
end;
