Let $(X,*)$ be a semigroup and $(G,+)$ a commutative fully ordered group. The absolute $|a|$ of an element $a\in G$ is defined as $|a|=\max(a,-a)$. extsc{Definition 1}. The function $f:X\to G$ with the following properties \begin{gather*} (U_1) f(x*y)eq f(x)+f(y); (U2) f(x*y)\geq|f(x)-f(y)| \end{gather*} for all $x,у\in X$ is called a general quasi-norm on $X$. The convergence of sequences in $X$ is induced by the function \[ d(x,y)=|f(x)-f(y)|\quad(d-im) \] (definition 3) using the convergence of sequences in $G$ (definition 2). Next we suppose that on $G$ there exists a root function $\gamma_2$, i.e., $\gamma_2:G\to G$ and hold that \begin{align*} &(A_1) \gamma_2(a+b)=\gamma_2(a)+\gamma_2(b) &(A_2) \gamma_2(a+a)=a \end{align*} for all $a,b\in G$. We define an equivalence relation on $X$ \[ xim y verset{peratorname{def}}eftrightarrow f(x)-f(y)=0 \] extsc{Diagonal theorem}. Let $x_{ij}\in X/_\sim(i,j\in N)$ and $d-\lim$ $x_{ij}=y$, $f(y)=0$, for $i=1,2,\ldots$ and there exists \[ f(\underset{jı K}*x_{ij})verset{peratorname{def}}=im_{noıfty}f(verset{n}{\underset{s=1}*}x_{ij_s}) \] for each $i\in N$ and each $K\subset N$, where $\{j_s\}$ is the increasing sequence of all elements of $K$. Then there exist an infinite set $I\subset N$ and a subset $J$ (finite or infinite) such that, for all $i\in I$, we have \[ f(\underset{jı J}*x_{ij})\geq\gamma(f(x_{ij})). \] Let $R$ be a $\sigma$-ring of sets. A multivalued set function $\mu:R\to X/_\sim$ is a member of the family $\mathcal F_0$ if it satisfies 1) $\mu(A\cup B)=\mu(A)*\mu(B)$ (set equality) for $A\cap B=\varnothing$ and $A,B\in R$; 2) $\displaystyle\lim_{k\to\infty}f(\overset{k}{\underset{n=1}*}\mu(E_n))=f(\mu(\overset{\infty}{\underset{n=1}\cup}e_n))$ for each disjoint sequence $\{E_n\}\subset R$; 3) $\displaystyle\lim_{n\to\infty}f(\mu(E_n))=0$ for each disjoint sequence $\{E_n\}\subset R$. extsc{Theorem}. Let $\mathcal F$ be a subfamily of the family $\mathcal F$ such that for some $g\in G(g>0)$ there exist $k_E\in N$ for all $E\in R$ such that \begin{align*} &f(\mu(E))<k_Eg\qquad ext{for}\quad\muı\mathcal F &(k_Eg=\underbrace{g+...+g}_{k_E}) \end{align*} Then there exists $r\in N$ such that \[ f(\mu(E))<rg\qquadext{for}\quad\muı\mathcal Fext{ and } Eı R. \]