be conic canonical relations. Define their fibre product over $T^*Y \setminus 0$ by
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\begin{align*}
F := \{(\rho_X,\rho_Y,\rho_Z) : (\rho_X,\rho_Y) \in C_1 \text{ and } (\rho_Y,\rho_Z) \in C_2\}.
\end{align*}
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Assume that this fibre product is clean of constant excess $e$, meaning that $F$ is a smooth conic manifold, every fibre of the projection $p:F \to p(F)$ has dimension $e$, and, at every point of $F$, its tangent space is the fibre product of $T C_1$ and $T C_2$ over $T(T^*Y \setminus 0)$.
and that $C_1 \circ C_2$ is a conic embedded Lagrangian submanifold of $(T^*X \setminus 0) \times (T^*Z \setminus 0)$ with respect to $\pi_X^*\omega_X - \pi_Z^*\omega_Z$.
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If $m_1,m_2 \in \mathbb{R}$, $A \in I^{m_1}(X,Y;C_1)$, and $B \in I^{m_2}(Y,Z;C_2)$ are properly supported Fourier integral operators acting on half-densities, then $AB$ is a well-defined properly supported Fourier integral operator and
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\begin{align*}
AB \in I^{m_1+m_2+e/2}(X,Z;C_1 \circ C_2).
\end{align*}
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Moreover, with the standard half-density principal-symbol convention, if
as a principal symbol on $C_1 \circ C_2$, modulo symbols of order $m_1+m_2+e/2-1$. Here $p_!$ denotes fibre integration of the induced half-density over the $e$-dimensional clean fibres of $p$, with the Maslov-bundle identification determined by the clean reduction.