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

theorem Th25:
  (sinh.p)^2 + (cosh.q)^2 = (cosh.(p+q))*(cosh.(p-q)) & (cosh.(p+q
))*(cosh.(p-q)) = (cosh.p)^2 + (sinh.q)^2 & (sinh.p)^2 + (cosh.q)^2 = (cosh.p)
  ^2 + (sinh.q)^2
proof
A1: (cosh.(p+q))*(cosh.(p-q)) =( (cosh.p)*(cosh.q) + (sinh.p)*(sinh.q) )* (
  cosh.(p + -q)) by Lm2
    .=( ( (cosh.p)*(cosh.q) ) + ( (sinh.p)*(sinh.q) ) )* ( ( (cosh.p)*(cosh.
  -q) ) + ( (sinh.p)*(sinh.-q) ) ) by Lm2
    .=( (cosh.p)*(cosh.q) )*( (cosh.p)*(cosh.-q) ) +( (cosh.p)*(cosh.q) )*(
(sinh.p)*(sinh.-q) ) +( (sinh.p)*(sinh.q) )*( (cosh.p)*(cosh.-q) ) +( (sinh.p)*
  (sinh.q) )*( (sinh.p)*(sinh.-q) )
    .=( (cosh.p)*(cosh.q) )*( (cosh.p)*(cosh.q) ) +( (cosh.p)*(cosh.q) )*( (
sinh.-q)*(sinh.p) ) +( (sinh.p)*(sinh.q) )*( (cosh.p)*(cosh.-q) ) +( (sinh.p)*(
  sinh.q) )*( (sinh.-q)*(sinh.p) ) by Th19
    .=( (cosh.p)*(cosh.q) )^2 +( (sinh.-q)*(sinh.p) )*( (cosh.p)*(cosh.q) )
+( (sinh.p)*(sinh.q) )*( (cosh.p)*(cosh.q) ) +( (sinh.-q)*(sinh.p) )*( (sinh.p)
  *(sinh.q) ) by Th19
    .=( (cosh.p)*(cosh.q) )^2 +( (-sinh.q)*(sinh.p) )*( (cosh.p)*(cosh.q) )
+( (sinh.p)*(sinh.q) )*( (cosh.p)*(cosh.q) ) +( (sinh.-q)*(sinh.p) )*( (sinh.p)
  *(sinh.q) ) by Th19
    .=( (cosh.p)*(cosh.q) )^2 + 0 +( (-1)*(sinh.q)*(sinh.p) )*( (sinh.p)*(
  sinh.q) ) by Th19
    .=(cosh.p)^2*(cosh.q)^2 - (sinh.q)^2*(sinh.p)^2;
  then
A2: (cosh.(p+q))*(cosh.(p-q)) =(cosh.p)^2*( (cosh.q)^2-(sinh.q)^2 ) + ((cosh
  .p)^2*(sinh.q)^2) + -((sinh.q)^2*(sinh.p)^2)
    .=(cosh.p)^2*1 + ((cosh.p)^2*(sinh.q)^2) + -((sinh.q)^2*(sinh.p)^2) by Th14
    .=(cosh.p)^2+(sinh.q)^2*((cosh.p)^2-(sinh.p)^2)
    .=(cosh.p)^2+(sinh.q)^2*1 by Th14
    .=(cosh.p)^2+(sinh.q)^2;
  (cosh.(p+q))*(cosh.(p-q)) =(cosh.q)^2*((cosh.p)^2-(sinh.p)^2) +( (cosh.q
  )^2*(sinh.p)^2 + -(sinh.p)^2*(sinh.q)^2 ) by A1
    .=(cosh.q)^2*1 +( (cosh.q)^2*(sinh.p)^2-(sinh.p)^2*(sinh.q)^2 ) by Th14
    .=(cosh.q)^2+(sinh.p)^2*( (cosh.q)^2-(sinh.q)^2 )
    .=(cosh.q)^2+(sinh.p)^2*1 by Th14
    .=(cosh.q)^2+(sinh.p)^2;
  hence thesis by A2;
end;
