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

<b>let </b><font color="Maroon">c<sub>2</sub></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="E1:141"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is <a href="limfunc1.html#V3" target="_self">convergent_in+infty</a> &amp;  <a href="limfunc1.html#K5" target="_self">lim_in+infty</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/>

<b>given </b><font color="Maroon">c<sub>3</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><a NAME="E2:141"/><i><font color="Green">E70</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4" target="_self">right_open_halfline</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/>

<b>thus </b><a NAME="E3:141"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a>ex <font color="Olive">b<sub>2</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rfunct_1.html#K4">^</a> </span>)</span> )
  <i><font color="Red">:: according to </font></i><a class="ref" href="limfunc1.html#D8" target="_self">LIMFUNC1:def 8</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="141_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E2:141_1"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="limfunc1.html#E1:141"><i><font color="Green">E69</font></i></a>, <a class="ref" href="limfunc1.html#D6" target="_self">Def6</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="141_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="141_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:141_1_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="141_1_1_1_1"><b>suppose </b></a><a NAME="E1:141_1_1_1_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:141_1_1_1_1"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="limfunc1.html#E1:141"><i><font color="Green">E69</font></i></a>, <a class="ref" href="limfunc1.html#D6" target="_self">Def6</a></i>;<br/><b>take </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font>;<br/><a NAME="E4:141_1_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="limfunc1.html#E1:141_1_1_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="limfunc1.html#E3:141_1_1_1_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E5:141_1_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="limfunc1.html#E2:141_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:141_1_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="limfunc1.html#E3:141_1_1_1_1"><i><font color="Green">E73</font></i></a></i>;<br/><a NAME="E7:141_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4" target="_self">right_open_halfline</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E8:141_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4" target="_self">right_open_halfline</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="limfunc1.html#E3:141_1_1_1_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E9:141_1_1_1_1"/><b>then </b>
0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="limfunc1.html#E2:141"><i><font color="Green">E70</font></i></a></i>;<br/><a NAME="E10:141_1_1_1_1"/><b>then </b>
not <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></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="E11:141_1_1_1_1"/><b>then </b>
not <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K11">"</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="E12:141_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K11">"</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="txt" href="limfunc1.html#E3:141_1_1_1_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E13:141_1_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="141_1_1_1_2"><b>suppose </b></a><a NAME="E1:141_1_1_1_2"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:141_1_1_1_2"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="limfunc1.html#E1:141"><i><font color="Green">E69</font></i></a>, <a class="ref" href="limfunc1.html#D6" target="_self">Def6</a></i>;<br/><b>take </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font>;<br/><b>thus </b><a NAME="E4:141_1_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="limfunc1.html#E2:141_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="limfunc1.html#E3:141_1_1_1_2"><i><font color="Green">E73</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:141_1_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="limfunc1.html#E1:141_1_1_1_2"><i><font color="Green">E72</font></i></a>, <a class="txt" href="limfunc1.html#E3:141_1_1_1_2"><i><font color="Green">E73</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:141_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 ;<br/><a NAME="E7:141_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4" target="_self">right_open_halfline</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E8:141_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4" target="_self">right_open_halfline</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="limfunc1.html#E3:141_1_1_1_2"><i><font color="Green">E73</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E9:141_1_1_1_2"/><b>then </b>
0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="limfunc1.html#E2:141"><i><font color="Green">E70</font></i></a></i>;<br/><a NAME="E10:141_1_1_1_2"/><b>then </b>
not <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></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="E11:141_1_1_1_2"/><b>then </b>
not <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K11">"</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="E12:141_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K11">"</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="txt" href="limfunc1.html#E3:141_1_1_1_2"><i><font color="Green">E73</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E13:141_1_1_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></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></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:141_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rfunct_1.html#K4">^</a> </span>)</span> )
 ;<br/>


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

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>

<b>assume </b><a NAME="E4:141"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> is <a href="limfunc1.html#V1" target="_self">divergent_to+infty</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rfunct_1.html#K4">^</a> </span>)</span> )
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:141"/><i><font color="Green">E72</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="limfunc1.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E7:141"/>
<font color="Maroon">c<sub>4</sub></font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">c<sub>5</sub></font> is    <a href="seqm_3.html#M1">subsequence</a> of <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="seqm_3.html#T47">SEQM_3:47</a></i>;<br/>
<a NAME="E8:141"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">c<sub>5</sub></font> is <a href="limfunc1.html#V1" target="_self">divergent_to+infty</a>
 
<b>by </b><i><a class="txt" href="limfunc1.html#E4:141"><i><font color="Green">E71</font></i></a>, <a class="ref" href="limfunc1.html#T53" target="_self">Th53</a></i>;<br/>
<a NAME="E9:141"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rfunct_1.html#K4">^</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K11">"</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="E10:141"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rfunct_1.html#K4">^</a> </span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<a NAME="E11:141"/><b>then </b><i><font color="Green">E74</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="limfunc1.html#E4:141"><i><font color="Green">E71</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E12:141"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="rfunct_2.html#T7">RFUNCT_2:7</a></i>;<br/>
<a NAME="E13:141"/><b>then </b><i><font color="Green">E75</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rfunct_1.html#K4">^</a> </span>)</span> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="limfunc1.html#E4:141"><i><font color="Green">E71</font></i></a>, <a class="txt" href="limfunc1.html#E11:141"><i><font color="Green">E74</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E14:141"/><b>then </b><i><font color="Green">E76</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="rfunct_2.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">c<sub>5</sub></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">c<sub>2</sub></font> <a href="rfunct_2.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="txt" href="limfunc1.html#E1:141"><i><font color="Green">E69</font></i></a>, <a class="txt" href="limfunc1.html#E8:141"><i><font color="Green">E73</font></i></a>, <a class="ref" href="limfunc1.html#D12" target="_self">Def12</a></i>;<br/>
<div><i><font color="Green">E77</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="141_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:141_2"/><i><font color="Green">E78</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T10">RFUNCT_2:10</a></i>;<br/><a NAME="E2:141_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E3:141_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="limfunc1.html#E6:141"><i><font color="Green">E72</font></i></a></i>;<br/><a NAME="E4:141_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="seqm_3.html#D9">SEQM_3:def 9</a></i>;<br/><a NAME="E5:141_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 ;<br/><a NAME="E6:141_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4" target="_self">right_open_halfline</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E7:141_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4" target="_self">right_open_halfline</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="limfunc1.html#E13:141"><i><font color="Green">E75</font></i></a>, <a class="txt" href="limfunc1.html#E1:141_2"><i><font color="Green">E78</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E8:141_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 <b>by </b><i><a class="txt" href="limfunc1.html#E2:141"><i><font color="Green">E70</font></i></a></i>;<br/><b>hence </b><a NAME="E9:141_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rfunct_2.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 <b>by </b><i><a class="txt" href="limfunc1.html#E13:141"><i><font color="Green">E75</font></i></a>, <a class="ref" href="rfunct_2.html#T21">RFUNCT_2:21</a></i>;<br/></div><b>end;</b></div>
<a NAME="E16:141"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rfunct_2.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 
;<br/>
<a NAME="E17:141"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="rfunct_2.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="seq_1.html#NR1" target="_self">is_not_0</a> 
 
<b>by </b><i><a class="ref" href="seq_1.html#T7">SEQ_1:7</a></i>;<br/>
<a NAME="E18:141"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rfunct_2.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 
<b>by </b><i><a class="txt" href="limfunc1.html#E15:141"><i><font color="Green">E77</font></i></a></i>;<br/>
<a NAME="E19:141"/><b>then </b><i><font color="Green">E79</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rfunct_2.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K18">"</a>  is <a href="limfunc1.html#V2" target="_self">divergent_to-infty</a>
 
<b>by </b><i><a class="txt" href="limfunc1.html#E14:141"><i><font color="Green">E76</font></i></a>, <a class="txt" href="limfunc1.html#E17:141"><i><font color="Green">E78</font></i></a>, <a class="ref" href="limfunc1.html#T63" target="_self">Th63</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rfunct_2.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">c<sub>5</sub></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">c<sub>2</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="seq_1.html#K18">"</a> 
<b>by </b><i><a class="txt" href="limfunc1.html#E11:141"><i><font color="Green">E74</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">c<sub>2</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">c<sub>5</sub></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">c<sub>2</sub></font> <a href="rfunct_1.html#K4">^</a> </span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="limfunc1.html#E4:141"><i><font color="Green">E71</font></i></a>, <a class="ref" href="rfunct_2.html#T27">RFUNCT_2:27</a></i>
;<br/>
<b>hence </b><a NAME="E21:141"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rfunct_1.html#K4">^</a> </span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font> is <a href="limfunc1.html#V2" target="_self">divergent_to-infty</a>
 <b>by </b><i><a class="txt" href="limfunc1.html#E19:141"><i><font color="Green">E79</font></i></a>, <a class="ref" href="limfunc1.html#T34" target="_self">Th34</a></i>;<br/>


</div>
