proof of homomorphic image of a $\\Sigma$ -structure is a $\\Sigma$ -structure

We need to show that $\\operatorname{im}(f)$ is closed under functions. For every constant symbol $c$ of $\\Sigma$ , $c^{\\mathfrak{B}}=f(c^{\\mathfrak{A}})$ . Hence $c^{\\mathfrak{B}}\\in\\operatorname{im}(f)$ . Also, if $b_{1},\\ldots,b_{n}\\in\\operatorname{im}(f)$ and $F$ is an $n$ -ary function symbol of $\\Sigma$ , then for some $a_{1},\\ldots,a_{n}\\in\\mathfrak{A}$ we have

F^{\\mathfrak{B}}(b_{1},\\ldots,b_{n})=F^{\\mathfrak{B}}(f(a_{1}),\\ldots,f(a_{n})%)=f(F^{\\mathfrak{A}}(a_{1},\\ldots,a_{n})).

Hence $F^{\\mathfrak{B}}(b_{1},\\ldots,b_{n})\\in\\operatorname{im}(f)$ .

proof of homomorphic image of a Σ-structure is a Σ-structure

proof of homomorphic image of a $\\Sigma$ -structure is a $\\Sigma$ -structure