Extension of continuous mappings and $H_1$-retracts

We prove that any continuous mapping $f:E\to Y$ on a completely metrizable subspace $E$ of a perfect paracompact space $X$ can be extended to a Lebesgue class one mapping $g:X\to Y$ (i.e. for every open set $V$ in $Y$ the preimage $g^{-1}(V)$ is an $F_σ$-set in $X$) with values in an arbitrary topological space $Y$.

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