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

theorem
  tanh.p + tanh.r = (sinh.(p+r))/((cosh.p)*(cosh.r)) & tanh.p - tanh.r =
  (sinh.(p-r))/((cosh.p)*(cosh.r))
proof
A1: (sinh.(p-r))/((cosh.p)*(cosh.r)) =( (sinh.p)*(cosh.r)-(cosh.p)*(sinh.r)
  )/((cosh.p)*(cosh.r)) by Lm8
    .= ((sinh.p)*(cosh.r))/((cosh.p)*(cosh.r)) -((cosh.p)*(sinh.r))/((cosh.p
  )*(cosh.r)) by XCMPLX_1:120
    .=(sinh.p)/(cosh.p)-((cosh.p)*(sinh.r))/((cosh.p)*(cosh.r)) by Th15,
XCMPLX_1:91
    .=(sinh.p)/(cosh.p)-(sinh.r)/(cosh.r) by Th15,XCMPLX_1:91
    .=tanh.p-(sinh.r)/(cosh.r) by Th17
    .=tanh.p-tanh.r by Th17;
  (sinh.(p+r))/((cosh.p)*(cosh.r)) =( (sinh.p)*(cosh.r)+(cosh.p)*(sinh.r)
  )/((cosh.p)*(cosh.r)) by Lm3
    .= ((sinh.p)*(cosh.r))/((cosh.p)*(cosh.r)) +((cosh.p)*(sinh.r))/((cosh.p
  )*(cosh.r)) by XCMPLX_1:62
    .=(sinh.p)/(cosh.p)+((cosh.p)*(sinh.r))/((cosh.p)*(cosh.r)) by Th15,
XCMPLX_1:91
    .=(sinh.p)/(cosh.p)+(sinh.r)/(cosh.r) by Th15,XCMPLX_1:91
    .=tanh.p+(sinh.r)/(cosh.r) by Th17
    .=tanh.p+tanh.r by Th17;
  hence thesis by A1;
end;
