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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">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="E1:35"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">f</font> <a href="limfunc3.html#R1" target="_self">is_convergent_in</a> <font color="Maroon">x0</font>
 ;<br/>

<a NAME="E2:35"/><i><font color="Green">E38</font></i>: 
 <a href="limfunc3.html#K1" target="_self">lim</a> <font color="Maroon">f</font>,<font color="Maroon">x0</font> <a href="hidden.html#R1">=</a>  <a href="limfunc3.html#K1" target="_self">lim</a> <font color="Maroon">f</font>,<font color="Maroon">x0</font>
 
;<br/>
<a NAME="E3:35"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">r1</font>, <font color="Olive">r2</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font> &amp; <font color="Maroon">x0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r2</font> holds <br/> ex <font color="Olive">g1</font>, <font color="Olive">g2</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">g1</font> &amp; <font color="Olive">g1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font> &amp; <font color="Olive">g1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Olive">g2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r2</font> &amp; <font color="Maroon">x0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">g2</font> &amp; <font color="Olive">g2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:35"/><b>then </b><i><font color="Green">E45</font></i>: 
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#R1">&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#R1">&lt;</a> <font color="Olive">g</font> &amp; <font color="Olive">g</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font> &amp; <font color="Olive">g</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> )
 
<b>by </b><i/>;<br/>
<div><i><font color="Green">E46</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_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="E1:35_1"/><i><font color="Green">E48</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="relat_1.html#K2">rng</a> <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">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> )
 ;<br/><a NAME="E2:35_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:35_1"/>
( <font color="Maroon">f</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">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="limfunc3.html#K1" target="_self">lim</a> <font color="Maroon">f</font>,<font color="Maroon">x0</font> )
 <b>by </b><i>, , <a class="txt" href="limfunc3.html#E1"><i><font color="Green">Lemma31</font></i></a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:35"/>
<font color="Maroon">f</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/>

<a NAME="E7:35"/><b>then </b><i><font color="Green">E49</font></i>: 
 <a href="limfunc2.html#K1">lim_left</a> <font color="Maroon">f</font>,<font color="Maroon">x0</font> <a href="hidden.html#R1">=</a>  <a href="limfunc3.html#K1" target="_self">lim</a> <font color="Maroon">f</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="E8:35"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Maroon">x0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r</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#R1">&lt;</a> <font color="Olive">r</font> &amp; <font color="Maroon">x0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">g</font> &amp; <font color="Olive">g</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> )
 
<b>by </b><i>, </i>;<br/>
<div><i><font color="Green">E51</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><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="E1:35_2"/><i><font color="Green">E52</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="relat_1.html#K2">rng</a> <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">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> )
 ;<br/><a NAME="E2:35_2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:35_2"/>
( <font color="Maroon">f</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">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="limfunc3.html#K1" target="_self">lim</a> <font color="Maroon">f</font>,<font color="Maroon">x0</font> )
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:35"/>
<font color="Maroon">f</font> <a href="limfunc2.html#R4">is_right_convergent_in</a> <font color="Maroon">x0</font>
 <b>by </b><i>, <a class="ref" href="limfunc2.html#D4">LIMFUNC2:def 4</a></i>;<br/>

<b>hence </b><a NAME="E11:35"/>
(  <a href="limfunc2.html#K1">lim_left</a> <font color="Maroon">f</font>,<font color="Maroon">x0</font> <a href="hidden.html#R1">=</a>  <a href="limfunc2.html#K2">lim_right</a> <font color="Maroon">f</font>,<font color="Maroon">x0</font> &amp;  <a href="limfunc3.html#K1" target="_self">lim</a> <font color="Maroon">f</font>,<font color="Maroon">x0</font> <a href="hidden.html#R1">=</a>  <a href="limfunc2.html#K1">lim_left</a> <font color="Maroon">f</font>,<font color="Maroon">x0</font> &amp;  <a href="limfunc3.html#K1" target="_self">lim</a> <font color="Maroon">f</font>,<font color="Maroon">x0</font> <a href="hidden.html#R1">=</a>  <a href="limfunc2.html#K2">lim_right</a> <font color="Maroon">f</font>,<font color="Maroon">x0</font> )
 <b>by </b><i><a class="txt" href="limfunc3.html#E1"><i><font color="Green">Lemma31</font></i></a>, , <a class="ref" href="limfunc2.html#D8">LIMFUNC2:def 8</a></i>;<br/>


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