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

theorem Th14:
  (cosh.p)^2-(sinh.p)^2=1 & (cosh.p)*(cosh.p)-(sinh.p)*(sinh.p)=1
proof
A1: (sinh.p)*(sinh.p) = ((exp_R.(p) - exp_R.(-p))/2)*(sinh.p) by Def1
    .= ((exp_R.(p) - exp_R.(-p))/2)*((exp_R.(p) - exp_R.(-p))/2) by Def1
    .= ((exp_R.(p))*(exp_R.(p))-(exp_R.(p))*(exp_R.(-p)) -(exp_R.(-p))*(
  exp_R.(p))+(exp_R.(-p))*(exp_R.(-p)))/(2*2)
    .= ((exp_R.(p+p))-(exp_R.(p))*(exp_R.(-p)) -(exp_R.(-p))*(exp_R.(p))+(
  exp_R.(-p))*(exp_R.(-p)))/4 by Th12
    .= ((exp_R.(p+p))-(exp_R.(p+-p)) -(exp_R.(-p))*(exp_R.(p))+(exp_R.(-p))*
  (exp_R.(-p)))/4 by Th12
    .= ((exp_R.(p+p))-(exp_R.(p+-p)) -(exp_R.(-p))*(exp_R.(p))+(exp_R.(-p+-p
  )))/4 by Th12
    .= ((exp_R.(p+p)) - 1 - 1 + (exp_R.(-p+-p)))/4 by Th12,Th13;
  (cosh.p)*(cosh.p) = ((exp_R.(p) + exp_R.(-p))/2)*(cosh.p) by Def3
    .= ((exp_R.(p) + exp_R.(-p))/2)*((exp_R.(p) + exp_R.(-p))/2) by Def3
    .= ((exp_R.(p))*(exp_R.(p))+(exp_R.(p))*(exp_R.(-p)) +(exp_R.(-p))*(
  exp_R.(p))+(exp_R.(-p))*(exp_R.(-p)))/(2*2)
    .= ((exp_R.(p+p))+(exp_R.(p))*(exp_R.(-p)) +(exp_R.(-p))*(exp_R.(p))+(
  exp_R.(-p))*(exp_R.(-p)))/4 by Th12
    .= ((exp_R.(p+p))+(exp_R.(p+-p)) +(exp_R.(-p))*(exp_R.(p))+(exp_R.(-p))*
  (exp_R.(-p)))/4 by Th12
    .= ((exp_R.(p+p))+(exp_R.(p+-p)) +(exp_R.(-p))*(exp_R.(p))+(exp_R.(-p+-p
  )))/4 by Th12
    .= ((exp_R.(p+p)) + 1 + 1 + (exp_R.(-p+-p)))/4 by Th12,Th13
    .= ((exp_R.(p+p)) + 2 + (exp_R.(-p+-p)))/4;
  hence thesis by A1;
end;
