<?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">r</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>thus </b><a NAME="E1:28"/>
( <font color="Maroon">f</font> <a href="limfunc2.html#R5" target="_self">is_right_divergent_to+infty_in</a> <font color="Maroon">x0</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 implies <font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font> <a href="limfunc2.html#R5" target="_self">is_right_divergent_to+infty_in</a> <font color="Maroon">x0</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">f</font> <a href="limfunc2.html#R5" target="_self">is_right_divergent_to+infty_in</a> <font color="Maroon">x0</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/>

<b>thus </b><a NAME="E2:28_1"/>
for <font color="Olive">r1</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Maroon">x0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r1</font> holds <br/> ex <font color="Olive">g</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r1</font> &amp; <font color="Maroon">x0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">g</font> &amp; <font color="Olive">g</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">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span> )
  <i><font color="Red">:: according to </font></i><a class="ref" href="limfunc2.html#D5" target="_self">LIMFUNC2:def 5</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_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>assume </b><a NAME="E1:28_1_1"/>
<font color="Maroon">x0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">g</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:28_1_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">x0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">g</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> )
 <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">g</font>
;<br/>

<b>thus </b><a NAME="E4:28_1_1"/>
( <font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">x0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">g</font> &amp; <font color="Maroon">g</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">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>;<br/>


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

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

<b>assume </b><a NAME="E3:28_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">seq</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">seq</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">seq</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">x0</font></span>)</span> )
 ;<br/>

<a NAME="E4:28_1"/><b>then </b><i><font color="Green">E39</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">seq</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">x0</font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>;<br/>
<a NAME="E5:28_1"/><b>then </b>
<font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">seq</font> is <a href="limfunc1.html#V1">divergent_to+infty</a>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E6:28_1"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">r</font> <a href="seq_1.html#K14">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">seq</font></span>)</span> is <a href="limfunc1.html#V1">divergent_to+infty</a>
 
<b>by </b><i>, <a class="ref" href="limfunc1.html#T40">LIMFUNC1:40</a></i>;<br/>
<a NAME="E7:28_1"/>
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">x0</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E8:28_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">seq</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E9:28_1"/>
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">seq</font> is <a href="limfunc1.html#V1">divergent_to+infty</a>
 <b>by </b><i>, <a class="ref" href="rfunct_2.html#T24">RFUNCT_2:24</a></i>;<br/>


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

<b>thus </b><a NAME="E2:28"/>
( <font color="Maroon">f</font> <a href="limfunc2.html#R5" target="_self">is_right_divergent_to+infty_in</a> <font color="Maroon">x0</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 implies <font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font> <a href="limfunc2.html#R6" target="_self">is_right_divergent_to-infty_in</a> <font color="Maroon">x0</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_2"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">f</font> <a href="limfunc2.html#R5" target="_self">is_right_divergent_to+infty_in</a> <font color="Maroon">x0</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 ;<br/>

<b>thus </b><a NAME="E2:28_2"/>
for <font color="Olive">r1</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Maroon">x0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r1</font> holds <br/> ex <font color="Olive">g</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r1</font> &amp; <font color="Maroon">x0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">g</font> &amp; <font color="Olive">g</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">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span> )
  <i><font color="Red">:: according to </font></i><a class="ref" href="limfunc2.html#D6" target="_self">LIMFUNC2:def 6</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_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>assume </b><a NAME="E1:28_2_1"/>
<font color="Maroon">x0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">g</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:28_2_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">x0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">g</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> )
 <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">g</font>
;<br/>

<b>thus </b><a NAME="E4:28_2_1"/>
( <font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">x0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">g</font> &amp; <font color="Maroon">g</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">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>;<br/>


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

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

<b>assume </b><a NAME="E3:28_2"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">seq</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">seq</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">seq</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">x0</font></span>)</span> )
 ;<br/>

<a NAME="E4:28_2"/><b>then </b><i><font color="Green">E52</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">seq</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">x0</font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>;<br/>
<a NAME="E5:28_2"/><b>then </b>
<font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">seq</font> is <a href="limfunc1.html#V1">divergent_to+infty</a>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E6:28_2"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">r</font> <a href="seq_1.html#K14">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">seq</font></span>)</span> is <a href="limfunc1.html#V2">divergent_to-infty</a>
 
<b>by </b><i>, <a class="ref" href="limfunc1.html#T40">LIMFUNC1:40</a></i>;<br/>
<a NAME="E7:28_2"/>
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">x0</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E8:28_2"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">seq</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E9:28_2"/>
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">seq</font> is <a href="limfunc1.html#V2">divergent_to-infty</a>
 <b>by </b><i>, <a class="ref" href="rfunct_2.html#T24">RFUNCT_2:24</a></i>;<br/>


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

<b>thus </b><a NAME="E3:28"/>
( <font color="Maroon">f</font> <a href="limfunc2.html#R6" target="_self">is_right_divergent_to-infty_in</a> <font color="Maroon">x0</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 implies <font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font> <a href="limfunc2.html#R6" target="_self">is_right_divergent_to-infty_in</a> <font color="Maroon">x0</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_3"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">f</font> <a href="limfunc2.html#R6" target="_self">is_right_divergent_to-infty_in</a> <font color="Maroon">x0</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/>

<b>thus </b><a NAME="E2:28_3"/>
for <font color="Olive">r1</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Maroon">x0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r1</font> holds <br/> ex <font color="Olive">g</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r1</font> &amp; <font color="Maroon">x0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">g</font> &amp; <font color="Olive">g</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">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span> )
  <i><font color="Red">:: according to </font></i><a class="ref" href="limfunc2.html#D6" target="_self">LIMFUNC2:def 6</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_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>assume </b><a NAME="E1:28_3_1"/>
<font color="Maroon">x0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">g</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:28_3_1"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">x0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">g</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> )
 <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">g</font>
;<br/>

<b>thus </b><a NAME="E4:28_3_1"/>
( <font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">x0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">g</font> &amp; <font color="Maroon">g</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">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>;<br/>


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

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

<b>assume </b><a NAME="E3:28_3"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">seq</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">seq</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">seq</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">x0</font></span>)</span> )
 ;<br/>

<a NAME="E4:28_3"/><b>then </b><i><font color="Green">E65</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">seq</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">x0</font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>;<br/>
<a NAME="E5:28_3"/><b>then </b>
<font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">seq</font> is <a href="limfunc1.html#V2">divergent_to-infty</a>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E6:28_3"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">r</font> <a href="seq_1.html#K14">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">seq</font></span>)</span> is <a href="limfunc1.html#V2">divergent_to-infty</a>
 
<b>by </b><i>, <a class="ref" href="limfunc1.html#T41">LIMFUNC1:41</a></i>;<br/>
<a NAME="E7:28_3"/>
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">x0</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E8:28_3"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">seq</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E9:28_3"/>
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">seq</font> is <a href="limfunc1.html#V2">divergent_to-infty</a>
 <b>by </b><i>, <a class="ref" href="rfunct_2.html#T24">RFUNCT_2:24</a></i>;<br/>


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

<b>assume </b><a NAME="E4:28"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">f</font> <a href="limfunc2.html#R6" target="_self">is_right_divergent_to-infty_in</a> <font color="Maroon">x0</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 ;<br/>

<b>thus </b><a NAME="E5:28"/>
for <font color="Olive">r1</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Maroon">x0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r1</font> holds <br/> ex <font color="Olive">g</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r1</font> &amp; <font color="Maroon">x0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">g</font> &amp; <font color="Olive">g</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">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span> )
  <i><font color="Red">:: according to </font></i><a class="ref" href="limfunc2.html#D5" target="_self">LIMFUNC2:def 5</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_4"><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>assume </b><a NAME="E1:28_4"/>
<font color="Maroon">x0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">g</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:28_4"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">x0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">g</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> )
 <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">g</font>
;<br/>

<b>thus </b><a NAME="E4:28_4"/>
( <font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">x0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">g</font> &amp; <font color="Maroon">g</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">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>;<br/>


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

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

<b>assume </b><a NAME="E6:28"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">seq</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">seq</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">seq</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">x0</font></span>)</span> )
 ;<br/>

<a NAME="E7:28"/><b>then </b><i><font color="Green">E70</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">seq</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">x0</font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>;<br/>
<a NAME="E8:28"/><b>then </b>
<font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">seq</font> is <a href="limfunc1.html#V2">divergent_to-infty</a>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E9:28"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">r</font> <a href="seq_1.html#K14">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">seq</font></span>)</span> is <a href="limfunc1.html#V1">divergent_to+infty</a>
 
<b>by </b><i>, <a class="ref" href="limfunc1.html#T41">LIMFUNC1:41</a></i>;<br/>
<a NAME="E10:28"/>
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">x0</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E11:28"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">seq</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E12:28"/>
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">seq</font> is <a href="limfunc1.html#V1">divergent_to+infty</a>
 <b>by </b><i>, <a class="ref" href="rfunct_2.html#T24">RFUNCT_2:24</a></i>;<br/>


</div>
