dots,a_n)=T\bar{f_i}(a_1,\ldots,a_n).$$

From the last equality we obtain $\bar{f_i}(a_1,\ldots,a_n)=0$, so $A\vDash\bar{f_i}$ and $\bar{f_i}\in T_0({\mathrm{Var}})$. By the previous proposition $f_i\in T_0({\mathrm{Di}}{\mathrm{Var}})$.

We recall that $f$ is called a multilinear s-identity (in the case of ordinary algebras) if $$f\in T_0({\mathcal{H}}{\mathrm{SJ}})\setminus T_0({\mathrm{Jord}}):={\mathrm{SId}}.$$

A similar notion can be introduced for dialgebras [@Br:09] $$f\in T_0({\mathcal{H}}{\mathrm{Di}}{\mathrm{SJ}})\setminus T_0({\mathrm{Di}}{\mathrm{Jord}}):={\mathrm{Di}}{\mathrm{SId}}.$$

\[thm:CorrespSId\]

1.  Let $g = g(x_1,\ldots,x_n)\in{\mathrm{SId}}$. Then $\Psi^{x_i}_{\mathrm{Alg}}\,g\in{\mathrm{Di}}{\mathrm{SId}}$ for all $i=1,\ldots,n$.

2.  Let $f=f(x_1,\ldots,x_n)\in{\mathrm{Di}}{\mathrm{SId}}$, $f=f_1+\ldots+f_n$ by a central letter. Then there exists $j\in\{1,\,\ldots,\,n\}$ such that $\bar{f_j}\in{\mathrm{SId}}$.

We prove the statement 1. Let $g\in{\mathrm{SId}}$, hence by the definition ${\mathrm{SId}}$ we have $g\in T_0({\mathcal{H}}{\mathrm{SJ}})$ and $g\not\in T_0({\mathrm{Jord}})$. Proposition \[prop:PsiDiVarVar\] implies $\Psi^{x_i}_{\mathrm{Alg}}\,g\in T_0({\mathrm{Di}}{\mathcal{H}}{\mathrm{SJ}})$, $\Psi^{x_i}_{\mathrm{Alg}}\,g\not\in
T_0({\mathrm{Di}}{\mathrm{Jord}})$. It follows from the equality of varieties ${\mathcal{H}}{\mathrm{Di}}{\mathrm{SJ}}={\mathrm{Di}}{\mathcal{H}}{\mathrm{SJ}}$ that $\Psi^{x_i}_{\mathrm{Alg}}\,g\in {\mathrm{Di}}{\mathrm{SId}}$.

For proving the statement 2 consider $f\in{\mathrm{Di}}{\mathrm{SId}}$. By the definition of ${\mathrm{Di}}{\mathrm{SId}}$ and Theorem \[thm:EqOfVarDialg\] we have $f\in T_0({\mathcal{H}}{\mathrm{Di}}{\mathrm{SJ}})=T_0({\mathrm{Di}}{\mathcal{H}}{\mathrm{SJ}})$ and $f\not\in
T_0({\mathrm{Di}}{\mathrm{Jord}})$. It follows from $f\in T_0({\mathrm{Di}}{\mathcal{H}}{\mathrm{SJ}})$ by Proposition \[prop:f1fnDiVarDiVar\] that $f_i\in T_0({\mathrm{Di}}{\mathcal{H}}{\mathrm{SJ}})$ for all $i$. It follows from $f\not\in T_0({\mathrm{Di