J. Mikusiński formulated the Diagonal theorem in paper [8]. P. Antosik [1], [2], generalized this theorem. The Diagonal theorem has many important applications in functional analysis and the measure theory. In this paper the authors prove some theorems in semi-goups and Banach spaces using the generalized Diagonal theorem of E. Pap [10]. extsc{Definition}. Let $(X,*)$ be a semi-goup. Let $f(x)$ be a real valued function on $X$ with following properties: \[ (F_1)f(x*y)eq f(x)+f(y),\quad (F_2)f(x*y)\geq|f(x)-f(y)|, \] then $X$ with the functional $f$ is a DT-space. The following theorem is a consequence of the Diagonal theorem [10]. extsc{Theorem 1}. \emph{Let $X$ be a DT-space and $а_{nk}\in Х$ $(п,k\in N)$ so that} \[ um_{n=1}^{ıfty}f(a_{nk}<ıfty)\qquad extit{for}\quad k=1,2,... \] \emph{If there exists a sequence of natural numbers $\{k_i\}i\in N$ and a sequence of subsets of the set $N\{S_i\}i\in N$ so that $\max_{s\in S_1} s<\min_{s\in S_{i+1}}s$, for $i=1,2,\ldots$ and $\varepsilon>0$ such that} \[ f(\underset{nı S_i}*a_{nk_{i}})>ǎrepsilon\qquadextit{for}\quad i=1,2,... \] \emph{then there exists an infinite set $I\in N$ and a set $S\subset N$ so that for every} $i\in I$ \[ f(\underset{nı S}*a_{nk_{i}})>\frac12ǎrepsilon\quad extit{holds}. \] /( * ank)>fiZ holds. ncs 2 Using theorem 1 we have the generalization of Shure lemma. extsc{Theorem 2}. \emph{Let $В$ be a Banach space with the positive Macphail constant $\mu(B)$, $a_{nk}\in B$ $(n,k\in N)$ and} \[ um_{n=1}^{ıfty}\|a_{nk}\|<ıfty\qquad extit{for}\quad k=1,2,... \] \emph{If for every subset $S\subset N$ is} \[ im_{koıfty}um_{nı S}a_{nk}=0, \] then \[ im_{koıfty}um_{nı N}\|a_{nk}\|=0. \] Using the results of J.\,B. Diaz and F.\,T. Metcalf [4] we have, in theorem 3, other generalization of Shure lemma. We have in theorem 4 the generalization of the convergence of sequences from space $l_1$ to $0$ (G. Köthe [6]): extsc{Theorem 4}. \emph{Let be $E$ a Banach space and $E'a$ the dual space of $E$. Let $a_{nk}\in E(n,k\in N)$ and} \[ um_{n=1}^{ıfty}\|a_{nk}\|<ıfty\qquadextit{for}\quad k=1,2,... \] \[ im_{koıfty}um_{n=1}^{ıfty}f_n(a_{nk})=0 \] \emph{for every $(f_1,f_2,\ldots)\in\prod_{1}^{\infty}E'$, $\|f_n\|_{E'}=0$ or $1$ and} \[ im_{koıfty}\|a_{nk}\|=0\qquadextit{for}\quad n=1,2,...,extit{ then} \] \[ im_{koıfty}um_{n=1}^{ıfty}\|a_{nk}\|=0. \] Theorem 5 is a integral version of theorem 1.