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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="E1:60"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">f1</font> <a href="limfunc2.html#R3">is_left_divergent_to-infty_in</a> <font color="Maroon">x0</font> &amp; <font color="Maroon">f2</font> is <a href="limfunc1.html#V6">convergent_in-infty</a> &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/>

<div><i><font color="Green">E27</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="60_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/><a NAME="E1:60_1"/><i><font color="Green">E30</font></i>: 
 <a href="limfunc1.html#K6">lim_in-infty</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#K6">lim_in-infty</a> <font color="Maroon">f2</font>
 ;<br/><b>assume </b><a NAME="E2:60_1"/><i><font color="Green">E33</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="E3:60_1"/><b>then </b><i><font color="Green">E35</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> <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="E4:60_1"/><b>then </b>
<font color="Maroon">f1</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">s</font> is <a href="limfunc1.html#V2">divergent_to-infty</a>
 <b>by </b><i>, , <a class="ref" href="limfunc2.html#D3">LIMFUNC2:def 3</a></i>;<br/><a NAME="E5:60_1"/><b>then </b>
( <font color="Maroon">f2</font> <a href="rfunct_2.html#K1">*</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> 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">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="hidden.html#R1">=</a>  <a href="limfunc1.html#K6">lim_in-infty</a> <font color="Maroon">f2</font> )
 <b>by </b><i>, , , <a class="ref" href="limfunc1.html#D13">LIMFUNC1:def 13</a></i>;<br/><b>hence </b><a NAME="E6:60_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="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">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="hidden.html#R1">=</a>  <a href="limfunc1.html#K6">lim_in-infty</a> <font color="Maroon">f2</font> )
 <b>by </b><i>, <a class="ref" href="rfunct_2.html#T39">RFUNCT_2:39</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:60"/>
<font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font> <a href="limfunc2.html#R1">is_left_convergent_in</a> <font color="Maroon">x0</font>
 <b>by </b><i>, <a class="ref" href="limfunc2.html#D1">LIMFUNC2:def 1</a></i>;<br/>

<b>hence </b><a NAME="E4:60"/>
 <a href="limfunc2.html#K1">lim_left</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>,<font color="Maroon">x0</font> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#K6">lim_in-infty</a> <font color="Maroon">f2</font>
 <b>by </b><i>, <a class="ref" href="limfunc2.html#D7">LIMFUNC2:def 7</a></i>;<br/>


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