reserve x, y, z, w for Real;
reserve n for Element of NAT;

theorem
  (cosh(x)+sinh x)/(cosh(x)-sinh(x)) = (1+tanh(x))/(1-tanh x)
proof
A1: exp_R(2*x) = (1+tanh(x))/(1-tanh((2*x)/2)) by Th7
    .= (1+tanh(x))/(1-tanh(x));
  exp_R(2*x) = (cosh((2*x)/2)+sinh((2*x)/2))/(cosh((2*x)/2)-sinh((2*x)/2))
  by Th7
    .= (cosh(x)+sinh(x))/(cosh(x)-sinh(x));
  hence thesis by A1;
end;
