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

theorem
  sinh.p + sinh.r = 2*(sinh.(p/2+r/2))*(cosh.(p/2-r/2)) & sinh.p - sinh.
  r = 2*(sinh.(p/2-r/2))*(cosh.(p/2+r/2))
proof
A1: 2*(sinh.(p/2-r/2))*(cosh.(p/2+r/2)) =2*( (sinh.(p/2))*(cosh.(r/2))-(cosh
  .(p/2))*(sinh.(r/2)) ) *(cosh.(p/2+r/2)) by Lm8
    .=2*((sinh.(p/2))*(cosh.(r/2))-(cosh.(p/2))*(sinh.(r/2))) *((cosh.(p/2))
  *(cosh.(r/2))+(sinh.(p/2))*(sinh.(r/2))) by Lm2
    .=2*( ((cosh.(r/2))*(sinh.(r/2)))*(-((cosh.(p/2))^2-(sinh.(p/2))^2)) +(
sinh.(p/2))*( (cosh.(p/2))*(cosh.(r/2))^2 ) -(cosh.(p/2))*( (sinh.(p/2))*(sinh.
  (r/2))^2 ) )
    .=2*( ((cosh.(r/2))*(sinh.(r/2)))*(-1) +(sinh.(p/2))*( (cosh.(p/2))*(
  cosh.(r/2))^2 ) -(cosh.(p/2))*( (sinh.(p/2))*(sinh.(r/2))^2 ) ) by Th14
    .=2*( ((sinh.(p/2))*(cosh.(p/2)))*((cosh.(r/2))^2-(sinh.(r/2))^2) +(-1)*
  ((cosh.(r/2))*(sinh.(r/2))) )
    .=2*( ((sinh.(p/2))*(cosh.(p/2)))*1 +(-1)*((cosh.(r/2))*(sinh.(r/2))) )
  by Th14
    .= 2*(sinh.(p/2))*(cosh.(p/2))-2*((sinh.(r/2))*(cosh.(r/2)))
    .=sinh.(2*(p/2)) - 2*(sinh.(r/2))*(cosh.(r/2)) by Lm6
    .=sinh.p - sinh.(2*(r/2)) by Lm6
    .=sinh.p - sinh.r;
  2*(sinh.(p/2+r/2))*(cosh.(p/2-r/2)) =2*((sinh.(p/2))*(cosh.(r/2))+(cosh.
  (p/2))*(sinh.(r/2))) *(cosh.(p/2-r/2)) by Lm3
    .=2*((sinh.(p/2))*(cosh.(r/2))+(cosh.(p/2))*(sinh.(r/2))) *((cosh.(p/2))
  *(cosh.(r/2))-(sinh.(p/2))*(sinh.(r/2))) by Lm7
    .=2*( ((sinh.(r/2))*(cosh.(r/2)))*((cosh.(p/2))^2-(sinh.(p/2))^2) +(sinh
.(p/2))*( (cosh.(p/2))*(cosh.(r/2))^2 ) -(cosh.(p/2))*( (sinh.(p/2))*(sinh.(r/2
  ))^2 ) )
    .=2*( ((sinh.(r/2))*(cosh.(r/2)))*1 +(sinh.(p/2))*( (cosh.(p/2))*(cosh.(
  r/2))^2 ) -(cosh.(p/2))*( (sinh.(p/2))*(sinh.(r/2))^2 ) ) by Th14
    .=2*( ((sinh.(p/2))*(cosh.(p/2)))*((cosh.(r/2))^2 -(sinh.(r/2))^2 ) +(
  sinh.(r/2))*(cosh.(r/2)) )
    .=2*( ((sinh.(p/2))*(cosh.(p/2))*1)+(sinh.(r/2))*(cosh.(r/2)) ) by Th14
    .=2*(sinh.(p/2))*(cosh.(p/2)) + 2*((sinh.(r/2))*(cosh.(r/2)))
    .=sinh.(2*(p/2)) + 2*(sinh.(r/2))*(cosh.(r/2)) by Lm6
    .=sinh.p + sinh.(2*(r/2)) by Lm6
    .=sinh.p + sinh.r;
  hence thesis by A1;
end;
