$n$-Body Phase Space Recurrence Relation

On slide 23 of these slides, it is stated that an $n$ body phase space element $dPhi_n(P; p_1, ldots, p_n)$ may be decomposed according to the recurrence relation

begin{align*}
mathrm{d} Phi_{n}left(P ; p_{1}, ldots, p_{n}right) &=mathrm{d} m_{12 ldots(n-1)}^{2} mathrm{~d} Phi_{2}left(P ; p_{12 ldots(n-1)}, p_{n}right)
&times mathrm{d} Phi_{n-1}left(P ; p_{1}, ldots, p_{(n-1)}right) tag{1}
end{align*}

where $P$ is the 4-momentum of the initial state with total mass $P^2 = M^2$ and

$$
m^2_{12ldots n} stackrel{text{def}}{=} (p_1 + p_2 +cdots +p_n)^2.
$$

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On the following slide, the result of $n=4$ is given by

begin{align*}
mathrm{d} Phi_{4}left(P ; p_{1}, p_{2}, p_{3}, p_{4}right) &propto frac{sqrt{lambdaleft(M^{2} ; m_{4}^{2}, m_{123}^{2}right)}}{M^{2}} m_{123} mathrm{~d} m_{123} mathrm{~d} Omega_{1234}
&times frac{sqrt{lambdaleft(m_{123}^{2} ; m_{3}^{2}, m_{12}^{2}right)}}{m_{123}^{2}} m_{12} mathrm{~d} m_{12} mathrm{~d} Omega_{123}
&times frac{sqrt{lambdaleft(m_{12}^{2} ; m_{1}^{2}, m_{2}^{2}right)}}{m_{12}^{2}} mathrm{~d} Omega_{12}tag{2}
end{align*}

where $lambda$ is the Källén fucnction and we have used the fact that

$$
mathrm{d} Phi_{2}left(P ; p_{1}, p_{2}right) propto frac{sqrt{lambdaleft(M^{2}, m_{1}^{2}, m_{2}^{2}right)}}{M^{2}} mathrm{~d} Omega_{12}
$$

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Question: I cannot see how equation (2) is derived from equation (1). It appears that the $n=4$ case is able to be reduced into a product of $dPhi_2(P; p_{123}, p_{4})$, $dPhi_2(p_{123}; p_{12}, p_{3})$ and
$dPhi_2(p_{12}; p_{1}, p_{2})$, but I don’t see how this can be if $dPhi_{n-1}$ always has $P$ as the first argument in (1). Is this simply a typo or am I missing something?