<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">x0</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">f1</font> be   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> ;<br/>





<b>assume </b><a NAME="E1:30"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">f1</font> <a href="limfunc3.html#R3" target="_self">is_divergent_to-infty_in</a> <font color="Maroon">x0</font> &amp; ( for <font color="Olive">r1</font>, <font color="Olive">r2</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font> &amp; <font color="Maroon">x0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r2</font> holds <br/> ex <font color="Olive">g1</font>, <font color="Olive">g2</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">g1</font> &amp; <font color="Olive">g1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font> &amp; <font color="Olive">g1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Olive">g2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r2</font> &amp; <font color="Maroon">x0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">g2</font> &amp; <font color="Olive">g2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> ) ) )
 ;<br/>

<b>given </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><a NAME="E2:30"/><i><font color="Green">E38</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> &amp; <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">x0</font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">x0</font>,<span class="p3">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">x0</font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">x0</font>,<span class="p3">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> &amp; ( for <font color="Olive">g</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">g</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">x0</font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">x0</font>,<span class="p3">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> holds <br/><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">g</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">g</font> ) )
 ;<br/>

<b>thus </b><a NAME="E3:30"/>
for <font color="Olive">r1</font>, <font color="Olive">r2</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font> &amp; <font color="Maroon">x0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r2</font> holds <br/> ex <font color="Olive">g1</font>, <font color="Olive">g2</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">g1</font> &amp; <font color="Olive">g1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font> &amp; <font color="Olive">g1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Olive">g2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r2</font> &amp; <font color="Maroon">x0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">g2</font> &amp; <font color="Olive">g2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> )
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="limfunc3.html#D3" target="_self">LIMFUNC3:def 3</a><br/>

<b>let </b><font color="Maroon">s</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>

<b>assume </b><a NAME="E4:30"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">s</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span> )
 ;<br/>

<b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:30"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/>( <font color="Maroon">x0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> &amp; <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E7:30"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">s</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font> )
 
<b>by </b><i>, <a class="ref" href="seq_4.html#T33">SEQ_4:33</a></i>;<br/>
<a NAME="E8:30"/><i><font color="Green">E48</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:30"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="rfunct_2.html#T7">RFUNCT_2:7</a></i>;<br/>
<a NAME="E10:30"/><b>then </b><i><font color="Green">E49</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:30_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:30_1"/><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T9">RFUNCT_2:9</a></i>;<br/><a NAME="E4:30_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E5:30_1"/><b>then </b>
( <font color="Maroon">x0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> &amp; <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r</font> )
 <b>by </b><i/>;<br/><a NAME="E6:30_1"/><b>then </b>
( <font color="Maroon">x0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r</font> )
 <b>by </b><i><a class="ref" href="seqm_3.html#D9">SEQM_3:def 9</a></i>;<br/><a NAME="E7:30_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g1</font> where <font color="Olive">g1</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">x0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">g1</font> &amp; <font color="Olive">g1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r</font> ) </span>}</span> 
 ;<br/><a NAME="E8:30_1"/><b>then </b><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D2">RCOMP_1:def 2</a></i>;<br/><a NAME="E9:30_1"/>
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T10">RFUNCT_2:10</a></i>;<br/><a NAME="E10:30_1"/><b>then </b>
not <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E11:30_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E12:30_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">x0</font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">x0</font>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i>, <a class="txt" href="limfunc3.html#E1"><i><font color="Green">Lemma31</font></i></a>, </i>;<br/></div><b>end;</b></div>
<a NAME="E12:30"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">x0</font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">x0</font>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E13:30"/><b>then </b><i><font color="Green">E52</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">x0</font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">x0</font>,<span class="p3">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/>
<a NAME="E14:30"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">x0</font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">x0</font>,<span class="p3">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E15:30"/><b>then </b><i><font color="Green">E65</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T18">XBOOLE_1:18</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:30_2"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span>
 ;<br/><a NAME="E2:30_2"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E3:30_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div>
<a NAME="E17:30"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E18:30"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">f1</font> <a href="rfunct_2.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> is <a href="limfunc1.html#V2">divergent_to-infty</a>
 
<b>by </b><i>, , </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:30_3"/>
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T10">RFUNCT_2:10</a></i>;<br/><a NAME="E2:30_3"/><b>then </b>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="limfunc3.html#E1"><i><font color="Green">Lemma31</font></i></a></i>;<br/><a NAME="E3:30_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="rfunct_2.html#T21">RFUNCT_2:21</a></i>;<br/><b>hence </b><a NAME="E4:30_3"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="rfunct_2.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="rfunct_2.html#T21">RFUNCT_2:21</a></i>;<br/></div><b>end;</b></div>
<a NAME="E20:30"/><b>then </b>
<font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> is <a href="limfunc1.html#V2">divergent_to-infty</a>
 
<b>by </b><i>, <a class="ref" href="limfunc1.html#T70">LIMFUNC1:70</a></i>;<br/>
<a NAME="E21:30"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">s</font></span>)</span> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font> is <a href="limfunc1.html#V2">divergent_to-infty</a>
 
<b>by </b><i>, <a class="ref" href="rfunct_2.html#T22">RFUNCT_2:22</a></i>;<br/>
<b>hence </b><a NAME="E22:30"/>
<font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">s</font> is <a href="limfunc1.html#V2">divergent_to-infty</a>
 <b>by </b><i><a class="ref" href="limfunc1.html#T34">LIMFUNC1:34</a></i>;<br/>


</div>
