<?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">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="E1:52"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">f</font> <a href="limfunc3.html#R1" target="_self">is_convergent_in</a> <font color="Maroon">x0</font> &amp;  <a href="limfunc3.html#K1" target="_self">lim</a> <font color="Maroon">f</font>,<font color="Maroon">x0</font> <a href="hidden.html#R1">=</a> 0 &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> &amp; <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">g1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">g2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 ) ) )
 ;<br/>

<b>given </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><a NAME="E2:52"/><i><font color="Green">E38</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> &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> 0 ) )
 ;<br/>

<b>thus </b><a NAME="E3:52"/>
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> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_1.html#K4">^</a> </span>)</span> &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> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_1.html#K4">^</a> </span>)</span> )
  <i><font color="Red">:: according to </font></i><a class="ref" href="limfunc3.html#D3" target="_self">LIMFUNC3:def 3</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r1</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">r2</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><a NAME="E1:52_1"/>
( <font color="Maroon">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="Maroon">r2</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">g1</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">g2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:52_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g1</font> &amp; <font color="Maroon">g1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font> &amp; <font color="Maroon">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="Maroon">g2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font> &amp; <font color="Maroon">x0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g2</font> &amp; <font color="Maroon">g2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">g1</font>
;<br/>

<b>take </b>
<font color="Maroon">g2</font>
;<br/>

<a NAME="E4:52_1"/>
( not <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> &amp; not <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:52_1"/><b>then </b>
( not <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> &amp; not <font color="Maroon">g2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E6:52_1"/><b>then </b>
( <font color="Maroon">g1</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#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span></span>)</span> &amp; <font color="Maroon">g2</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#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span></span>)</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="E7:52_1"/>
( <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g1</font> &amp; <font color="Maroon">g1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font> &amp; <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_1.html#K4">^</a> </span>)</span> &amp; <font color="Maroon">g2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font> &amp; <font color="Maroon">x0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g2</font> &amp; <font color="Maroon">g2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_1.html#K4">^</a> </span>)</span> )
 <b>by </b><i>, <a class="ref" href="rfunct_1.html#D8">RFUNCT_1:def 8</a></i>;<br/>


</div><b>end;</b></div>

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

<b>assume </b><a NAME="E4:52"/><i><font color="Green">E45</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> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_1.html#K4">^</a> </span>)</span></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="E6:52"/><i><font color="Green">E46</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="E7:52"/><i><font color="Green">E48</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="E8:52"/><i><font color="Green">E49</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> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_1.html#K4">^</a> </span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:52"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_1.html#K4">^</a> </span>)</span> <a href="hidden.html#R1">=</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">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#D8">RFUNCT_1:def 8</a></i>;<br/>
<a NAME="E10:52"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_1.html#K4">^</a> </span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 
;<br/>
<a NAME="E11:52"/><b>then </b><i><font color="Green">E50</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><a class="txt" href="limfunc3.html#E3"><i><font color="Green">Lemma37</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E12:52"/>
 <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="E13:52"/><b>then </b><i><font color="Green">E51</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> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_1.html#K4">^</a> </span>)</span></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> &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>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_1.html#K4">^</a> </span>)</span> &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>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> )
 
<b>by </b><i>, <a class="txt" href="limfunc3.html#E3"><i><font color="Green">Lemma37</font></i></a>, <a class="txt" href="limfunc3.html#E2:52"><i><font color="Green">E38</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E14:52"/>
 <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:52_2"/><i><font color="Green">E52</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="E2:52_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="E3:52_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">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="txt" href="limfunc3.html#E4"><i><font color="Green">Lemma40</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:52"/><b>then </b><i><font color="Green">E65</font></i>: 
( <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="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <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="hidden.html#R1">=</a> 0 )
 
<b>by </b><i>, <a class="txt" href="limfunc3.html#E1:52"><i><font color="Green">E36</font></i></a>, </i>;<br/>
<a NAME="E16:52"/><i><font color="Green">E66</font></i>: 
<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> <a href="seq_1.html#NR1" target="_self">is_not_0</a> 
 
<b>by </b><i>, <a class="ref" href="rfunct_2.html#T26">RFUNCT_2:26</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_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="E1:52_3"/><i><font color="Green">E67</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>  <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="E2:52_3"/>
<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="E3:52_3"/><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><a class="txt" href="limfunc3.html#E2"><i><font color="Green">Lemma35</font></i></a></i>;<br/><a NAME="E4:52_3"/><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="E5:52_3"/><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="E6:52_3"/><b>then </b><i><font color="Green">E68</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="E7:52_3"/>
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="txt" href="limfunc3.html#E6"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E8:52_3"/><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="limfunc3.html#T1" target="_self">Th1</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E9:52_3"/><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>,<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>, </i>;<br/><a NAME="E10:52_3"/><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"><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="txt" href="limfunc3.html#E6"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E11:52_3"/><b>then </b><i><font color="Green">E69</font></i>: 
<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> 0
 <b>by </b><i/>;<br/><a NAME="E12:52_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="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="limfunc3.html#E5"><i><font color="Green">Lemma41</font></i></a>, <a class="ref" href="seq_1.html#T7">SEQ_1:7</a></i>;<br/><b>hence </b><a NAME="E13:52_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> 0
 <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="E18:52"/><b>then </b>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/><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="Olive">n</font> <a href="xxreal_0.html#R1">&lt;</a> 0
 
;<br/>
<a NAME="E19:52"/><b>then </b><i><font color="Green">E70</font></i>: 
<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#K18">"</a>  is <a href="limfunc1.html#V2">divergent_to-infty</a>
 
<b>by </b><i><a class="txt" href="limfunc3.html#E4"><i><font color="Green">Lemma40</font></i></a>, <a class="txt" href="limfunc3.html#E5"><i><font color="Green">Lemma41</font></i></a>, <a class="ref" href="limfunc1.html#T63">LIMFUNC1:63</a></i>;<br/>
<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#K18">"</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<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></span>)</span> <a href="seq_1.html#K18">"</a> 
<b>by </b><i><a class="txt" href="limfunc3.html#E2:52"><i><font color="Green">E38</font></i></a>, <a class="ref" href="rfunct_2.html#T22">RFUNCT_2:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">s</font></span>)</span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="seqm_3.html#T41">SEQM_3:41</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_1.html#K4">^</a> </span>)</span> <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>
<b>by </b><i><a class="txt" href="limfunc3.html#E3"><i><font color="Green">Lemma37</font></i></a>, <a class="ref" href="rfunct_2.html#T27">RFUNCT_2:27</a></i>
;<br/>
<b>hence </b><a NAME="E21:52"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_1.html#K4">^</a> </span>)</span> <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="txt" href="limfunc3.html#E6"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="limfunc1.html#T34">LIMFUNC1:34</a></i>;<br/>


</div>
