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We study the theory of Lagrangian specialisation for such $f$, and prove a Gonz\\'{a}lez-Sprinberg type formula for the local Euler obstruction relative to $f$. With the help of this formula and MacPherson's graph construction for the vector bundle map $f^*T^*N \\to T^*M$, we find the Lagrangian cycle of the Milnor number constructible function $\\mu$. 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