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

theorem
  cosh.p + cosh.r = 2*(cosh.(p/2+r/2))*(cosh.(p/2-r/2)) & cosh.p - cosh.
  r = 2*(sinh.(p/2-r/2))*(sinh.(p/2+r/2))
proof
A1: 2*(sinh.(p/2-r/2))*(sinh.(p/2+r/2)) =2*((sinh.(p/2-r/2))*(sinh.(p/2+r/2) ))
    .=2*( (cosh.(p/2))^2 - (cosh.(r/2))^2 ) by Th24
    .=2*( 1/2*(cosh.(2*(p/2)) + 1)-(cosh.(r/2))^2 ) by Th18
    .=2*( 1/2*(cosh.p + 1)-1/2*(cosh.(2*(r/2)) + 1) ) by Th18
    .=cosh.p-cosh.r;
  2*(cosh.(p/2+r/2))*(cosh.(p/2-r/2)) =2*((cosh.(p/2+r/2))*(cosh.(p/2-r/2) ))
    .=2*( (sinh.(p/2))^2+(cosh.(r/2))^2 ) by Th25
    .=2*( 1/2*(cosh.(2*(p/2)) - 1)+(cosh.(r/2))^2 ) by Th18
    .=2*( 1/2*(cosh.p - 1)+1/2*(cosh.(2*(r/2)) + 1) ) by Th18
    .=cosh.r+cosh.p;
  hence thesis by A1;
end;
