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

theorem Th24:
  (sinh.p)^2 - (sinh.q)^2 = (sinh.(p+q))*(sinh.(p-q)) & (sinh.(p+q
))*(sinh.(p-q)) = (cosh.p)^2 - (cosh.q)^2 & (sinh.p)^2 - (sinh.q)^2 = (cosh.p)
  ^2 - (cosh.q)^2
proof
A1: (sinh.(p+q))*(sinh.(p-q)) =((sinh.p)*(cosh.q) + (cosh.p)*(sinh.q))*(sinh
  .(p+(-q))) by Lm3
    .=((sinh.p)*(cosh.q) + (cosh.p)*(sinh.q)) *((sinh.p)*(cosh.(-q)) + (cosh
  .p)*(sinh.(-q))) by Lm3
    .=((sinh.p)*(cosh.q) + (cosh.p)*(sinh.q)) *((sinh.p)*(cosh.q) + (cosh.p)
  *(sinh.(-q))) by Th19
    .=((sinh.p)*(cosh.q) + (cosh.p)*(sinh.q)) *((sinh.p)*(cosh.q) + (cosh.p)
  *(-sinh.q)) by Th19
    .= (sinh.p)^2*(cosh.q)^2-(sinh.q)^2*(cosh.p)^2;
  then
A2: (sinh.(p+q))*(sinh.(p-q)) =(cosh.q)^2*( -((cosh.p)^2-(sinh.p)^2) ) +( ((
  cosh.p)^2*(cosh.q)^2)+ -(sinh.q)^2*(cosh.p)^2 )
    .=(cosh.q)^2*( -1 ) +( (cosh.p)^2*( (cosh.q)^2-(sinh.q)^2 ) ) by Th14
    .=(cosh.q)^2*( -1 )+ ((cosh.p)^2*1) by Th14
    .=(cosh.p)^2 -(cosh.q)^2;
  (sinh.(p+q))*(sinh.(p-q)) =(sinh.p)^2*( (cosh.q)^2 -(sinh.q)^2 ) + ((
  sinh.q)^2*(sinh.p)^2)-(sinh.q)^2*(cosh.p)^2 by A1
    .=(sinh.p)^2*1 + (sinh.p)^2*(sinh.q)^2-(sinh.q)^2*(cosh.p)^2 by Th14
    .=(sinh.p)^2 + (sinh.q)^2*(-((cosh.p)^2-(sinh.p)^2))
    .=(sinh.p)^2 + (sinh.q)^2*(-1) by Th14
    .=(sinh.p)^2 - (sinh.q)^2;
  hence thesis by A2;
end;
