<?xml version="1.0"?>
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<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">f2</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="E1:24"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">f1</font> <a href="limfunc2.html#R1">is_left_convergent_in</a> <font color="Maroon">x0</font> &amp; <font color="Maroon">f2</font> <a href="limfunc2.html#R6">is_right_divergent_to-infty_in</a>  <a href="limfunc2.html#K1">lim_left</a> <font color="Maroon">f1</font>,<font color="Maroon">x0</font> &amp; ( for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x0</font> holds <br/> ex <font color="Olive">g</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">g</font> &amp; <font color="Olive">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x0</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">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font></span>)</span> ) ) )
 ;<br/>

<b>given </b><font color="Maroon">g</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><a NAME="E2:24"/><i><font color="Green">E27</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">g</font> &amp; ( for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">r</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">f1</font></span>)</span> <a href="subset_1.html#K9">/\</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">g</font></span>)</span>,<font color="Maroon">x0</font></span><a href="rcomp_1.html#K2">.[</a></span> holds <br/> <a href="limfunc2.html#K1">lim_left</a> <font color="Maroon">f1</font>,<font color="Maroon">x0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">r</font> ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><b>assume </b><a NAME="E1:24_1"/><i><font color="Green">E30</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="relset_1.html#K5">rng</a> <font color="Maroon">s</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">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">x0</font></span>)</span> )
 ;<br/><a NAME="E2:24_1"/><b>then </b><i><font color="Green">E33</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</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">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font></span>)</span> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a>  <a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">x0</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</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">f1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">x0</font></span>)</span> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">s</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font> )
 <b>by </b><i/>;<br/><a NAME="E3:24_1"/><b>then </b><i><font color="Green">E35</font></i>: 
( <font color="Maroon">f1</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">s</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">f1</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="limfunc2.html#K1">lim_left</a> <font color="Maroon">f1</font>,<font color="Maroon">x0</font> )
 <b>by </b><i>, , <a class="ref" href="limfunc2.html#D7">LIMFUNC2:def 7</a></i>;<br/><a NAME="E4:24_1"/>
<font color="Maroon">x0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="seq_2.html#K2">lim</a> <font color="Maroon">s</font>
 <b>by </b><i>, , </i>;<br/><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:24_1"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">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">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font>
 <b>by </b><i>, <a class="ref" href="limfunc2.html#T1">LIMFUNC2:1</a></i>;<br/><b>set </b><font color="Maroon">q</font> = <span class="p1">(<span class="default"><font color="Maroon">f1</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>;<br/><a NAME="E7:24_1"/><i><font color="Green">E39</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f1</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="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</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="hidden.html#R1">=</a>  <a href="limfunc2.html#K1">lim_left</a> <font color="Maroon">f1</font>,<font color="Maroon">x0</font> )
 <b>by </b><i>, <a class="ref" href="seq_4.html#T33">SEQ_4:33</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_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="E1:24_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</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>
 ;<br/><b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:24_1_1"/><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</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#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="E4:24_1_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="E5:24_1_1"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">x0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&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>
 <b>by </b><i/>;<br/><a NAME="E6:24_1_1"/><i><font color="Green">E42</font></i>: 
<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="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T10">RFUNCT_2:10</a></i>;<br/><a NAME="E7:24_1_1"/><b>then </b>
<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="hidden.html#R2">in</a>  <a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">x0</font>
 <b>by </b><i/>;<br/><a NAME="E8:24_1_1"/><b>then </b>
<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="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="Olive">g1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x0</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="prob_1.html#D15">PROB_1:def 15</a></i>;<br/><a NAME="E9:24_1_1"/><b>then </b>
 ex <font color="Olive">g1</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">g1</font> <a href="hidden.html#R1">=</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="Olive">g1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x0</font> )
 ;<br/><a NAME="E10:24_1_1"/><b>then </b>
<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="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g2</font> where <font color="Olive">g2</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">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">g2</font> &amp; <font color="Olive">g2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x0</font> ) </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E11:24_1_1"/><b>then </b>
<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="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">g</font></span>)</span>,<font color="Maroon">x0</font></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="E12:24_1_1"/><b>then </b>
<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="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font></span>)</span> <a href="subset_1.html#K9">/\</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">g</font></span>)</span>,<font color="Maroon">x0</font></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E13:24_1_1"/><b>then </b>
 <a href="limfunc2.html#K1">lim_left</a> <font color="Maroon">f1</font>,<font color="Maroon">x0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E14:24_1_1"/><b>then </b>
<font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r1</font> where <font color="Olive">r1</font> is   <a href="real_1.html#NM1">Real</a> :  <a href="limfunc2.html#K1">lim_left</a> <font color="Maroon">f1</font>,<font color="Maroon">x0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r1</font> </span>}</span> 
 ;<br/><a NAME="E15:24_1_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> <span class="p1">(<span class="default"><a href="limfunc2.html#K1">lim_left</a> <font color="Maroon">f1</font>,<font color="Maroon">x0</font></span>)</span>
 <b>by </b><i><a class="ref" href="limfunc1.html#D3">LIMFUNC1:def 3</a></i>;<br/><a NAME="E16:24_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">s</font></span>)</span> <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>
<b>by </b><i>, <a class="ref" href="rfunct_2.html#T21">RFUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="seqm_3.html#D9">SEQM_3:def 9</a></i>
;<br/><b>hence </b><a NAME="E18:24_1_1"/>
<font color="Maroon">x</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">f2</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> <span class="p2">(<span class="default"><a href="limfunc2.html#K1">lim_left</a> <font color="Maroon">f1</font>,<font color="Maroon">x0</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><a NAME="E9:24_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</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="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</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> <span class="p2">(<span class="default"><a href="limfunc2.html#K1">lim_left</a> <font color="Maroon">f1</font>,<font color="Maroon">x0</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E10:24_1"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">f2</font> <a href="rfunct_2.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</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> is <a href="limfunc1.html#V2">divergent_to-infty</a>
 <b>by </b><i>, , <a class="ref" href="limfunc2.html#D6">LIMFUNC2:def 6</a></i>;<br/><font color="Maroon">f2</font> <a href="rfunct_2.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</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> = 
<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="rfunct_2.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font>
<b>by </b><i>, <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">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font></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="ref" href="rfunct_2.html#T39">RFUNCT_2:39</a></i>
;<br/><b>hence </b><a NAME="E12:24_1"/>
<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font></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="ref" href="limfunc1.html#T34">LIMFUNC1:34</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:24"/>
<font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font> <a href="limfunc2.html#R3">is_left_divergent_to-infty_in</a> <font color="Maroon">x0</font>
 <b>by </b><i>, <a class="ref" href="limfunc2.html#D3">LIMFUNC2:def 3</a></i>;<br/>


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