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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">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>assume </b><a NAME="E1:51"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">f</font> <a href="limfunc2.html#R1" target="_self">is_left_convergent_in</a> <font color="Maroon">x0</font>
 ;<br/>

<div><i><font color="Green">E33</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><b>now </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="E1:51_1"/>
<font color="Maroon">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x0</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="E3:51_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">g</font> &amp; <font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x0</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><a class="txt" href="limfunc2.html#E1"><i><font color="Green">Lemma29</font></i></a>, </i>;<br/><b>take </b><font color="Maroon">g</font> = <font color="Maroon">g</font>;<br/><b>thus </b><a NAME="E4:51_1"/>
( <font color="Maroon">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">g</font> &amp; <font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x0</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="txt" href="limfunc2.html#E2"><i><font color="Green">Lemma31</font></i></a>, <a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E39</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">seq</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><a NAME="E1:51_2"/><i><font color="Green">E40</font></i>: 
 <a href="limfunc2.html#K1" target="_self">lim_left</a> <font color="Maroon">f</font>,<font color="Maroon">x0</font> <a href="hidden.html#R1">=</a>  <a href="limfunc2.html#K1" target="_self">lim_left</a> <font color="Maroon">f</font>,<font color="Maroon">x0</font>
 ;<br/><b>assume </b><a NAME="E2:51_2"/><i><font color="Green">E43</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#NK2" target="_self">left_open_halfline</a> <font color="Maroon">x0</font></span>)</span> )
 ;<br/><a NAME="E3:51_2"/><b>then </b><i><font color="Green">E44</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#NK2" target="_self">left_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="E4:51_2"/><b>then </b><i><font color="Green">E51</font></i>: 
( <font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">seq</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">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">seq</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="limfunc2.html#K1" target="_self">lim_left</a> <font color="Maroon">f</font>,<font color="Maroon">x0</font> )
 <b>by </b><i><a class="txt" href="limfunc2.html#E1"><i><font color="Green">Lemma29</font></i></a>, , <a class="txt" href="limfunc2.html#E2:51"><i><font color="Green">E33</font></i></a>, </i>;<br/><a NAME="E5:51_2"/><b>then </b><i><font color="Green">E52</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="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="ref" href="seq_2.html#T21">SEQ_2:21</a></i>;<br/><a NAME="E6:51_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#NK2" target="_self">left_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="E7:51_2"/><b>then </b><i><font color="Green">E53</font></i>: 
 <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="txt" href="limfunc2.html#E3"><i><font color="Green">Lemma34</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E8:51_2"/><b>then </b><i><font color="Green">E55</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> <a href="hidden.html#R1">=</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> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">seq</font>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T24">RFUNCT_2:24</a></i>;<br/><b>thus </b><a NAME="E9:51_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="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="ref" href="limfunc2.html#T1" target="_self">Th1</a>, , <a class="ref" href="rfunct_2.html#T24">RFUNCT_2:24</a></i>;<br/><b>thus </b><a NAME="E10:51_2"/>
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><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> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">seq</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="limfunc2.html#K1" target="_self">lim_left</a> <font color="Maroon">f</font>,<font color="Maroon">x0</font></span>)</span>
 <b>by </b><i><a class="txt" href="limfunc2.html#E4"><i><font color="Green">Lemma35</font></i></a>, , <a class="ref" href="seq_2.html#T22">SEQ_2:22</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:51"/>
<font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font> <a href="limfunc2.html#R1" target="_self">is_left_convergent_in</a> <font color="Maroon">x0</font>
 <b>by </b><i><a class="txt" href="limfunc2.html#E1:51"><i><font color="Green">E30</font></i></a>, </i>;<br/>

<b>hence </b><a NAME="E5:51"/>
 <a href="limfunc2.html#K1" target="_self">lim_left</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>,<font color="Maroon">x0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="limfunc2.html#K1" target="_self">lim_left</a> <font color="Maroon">f</font>,<font color="Maroon">x0</font></span>)</span>
 <b>by </b><i><a class="txt" href="limfunc2.html#E2"><i><font color="Green">Lemma31</font></i></a>, </i>;<br/>


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