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

theorem
  sinh.(2*p) = 2*(sinh.p)*(cosh.p) & cosh.(2*p) = 2*(cosh.p)^2 - 1 &
  tanh.(2*p) = (2*tanh.p)/(1+(tanh.p)^2)
proof
  tanh.(2*p)=tanh.(p+p) .=(tanh.p + tanh.p)/(1+ (tanh.p)*(tanh.p)) by Lm5
    .=(2*tanh.p)/(1+ (tanh.p)^2);
  hence thesis by Lm6;
end;
