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# multiplicity under specialization

Let $C$ be a projective non-singular curve defined over a field $K$ with the characteristic zero. Let $y,z$ be non-constant rational functions defined $K$ such that $y$ is defined at all poles and zeros of $z$ and gives an injective mapping of this set into the algebraic closure of $K.$ Let $(y_1,y_2,...,y_m)$ be all conjugates of $y$ over $K(z)$ and $r$ be the largest of the multiplicities of the zeros of $z.$ If $(y'_1,...,y'_m)$ is a specialization of $(y_1,y_2,...,y_m)$ over $z \rightarrow 0.$ \ Is this true that the multiplicity of any $y'_k$ in $(y'_1,...,y'_m)$ is less than $r ?$
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