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<b>let </b><font color="Maroon" title="c1">x0</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c2">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> , <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>



<b>assume </b><a NAME="E1:52"/><i><font color="Green" title="E30">A1</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="limfunc2.html#R1" title="LIMFUNC2:pred.1">is_left_convergent_in</a> <font color="Maroon" title="c1">x0</font>
 ;<br/>

<a NAME="E2:52"/><i><font color="Green" title="E31">A2</font></i>: 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="seq_1.html#K16" title="SEQ_1:func.16">-</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#T38" title="RFUNCT_1:th.38">RFUNCT_1:38</a></i>;<br/>
<b>hence </b><a NAME="E3:52"/>
 <a href="seq_1.html#K16" title="SEQ_1:func.16">-</a> <font color="Maroon" title="c2">f</font> <a href="limfunc2.html#R1" title="LIMFUNC2:pred.1">is_left_convergent_in</a> <font color="Maroon" title="c1">x0</font>
 <b>by </b><i><a class="txt" href="limfunc2.html#E1:52"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="limfunc2.html#T51" target="_self" title="LIMFUNC2:th.51">Th51</a></i>;<br/>

<b>thus </b> <a href="limfunc2.html#K1" title="LIMFUNC2:func.1">lim_left</a> <span class="p1">(<span class="default"><a href="seq_1.html#K16" title="SEQ_1:func.16">-</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c1">x0</font> = 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="limfunc2.html#K1" title="LIMFUNC2:func.1">lim_left</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">x0</font></span>)</span>
<b>by </b><i><a class="txt" href="limfunc2.html#E1:52"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="limfunc2.html#E2:52"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="limfunc2.html#T51" target="_self" title="LIMFUNC2:th.51">Th51</a></i>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="limfunc2.html#K1" title="LIMFUNC2:func.1">lim_left</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">x0</font></span>)</span>

;<br/>


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