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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="E1:45"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">f1</font> <a href="limfunc3.html#R1" target="_self">is_convergent_in</a> <font color="Maroon">x0</font> &amp; <font color="Maroon">f2</font> <a href="limfunc3.html#R1" target="_self">is_convergent_in</a> <font color="Maroon">x0</font> &amp;  <a href="limfunc3.html#K1" target="_self">lim</a> <font color="Maroon">f2</font>,<font color="Maroon">x0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; ( 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> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="rfunct_1.html#K2">/</a> <font color="Maroon">f2</font></span>)</span> &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> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="rfunct_1.html#K2">/</a> <font color="Maroon">f2</font></span>)</span> ) ) )
 ;<br/>

<a NAME="E2:45"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">f1</font> <a href="rfunct_1.html#K2">/</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="seq_1.html#K8">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="rfunct_1.html#K4">^</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#T47">RFUNCT_1:47</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_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>let </b><font color="Maroon">r2</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:45_1"/>
( <font color="Maroon">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="Maroon">r2</font> )
 ;<br/><b>then consider </b><font color="Maroon">g1</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">g2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:45_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g1</font> &amp; <font color="Maroon">g1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font> &amp; <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="rfunct_1.html#K2">/</a> <font color="Maroon">f2</font></span>)</span> &amp; <font color="Maroon">g2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font> &amp; <font color="Maroon">x0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g2</font> &amp; <font color="Maroon">g2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="rfunct_1.html#K2">/</a> <font color="Maroon">f2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="limfunc3.html#E1"><i><font color="Green">Lemma31</font></i></a></i>;<br/><b>take </b><font color="Maroon">g1</font> = <font color="Maroon">g1</font>;<br/><b>take </b><font color="Maroon">g2</font> = <font color="Maroon">g2</font>;<br/><b>thus </b><a NAME="E4:45_1"/>
( <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g1</font> &amp; <font color="Maroon">g1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font> )
 <b>by </b><i/>;<br/><a NAME="E5:45_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="rfunct_1.html#K2">/</a> <font color="Maroon">f2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="relat_1.html#K10">"</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rfunct_1.html#D4">RFUNCT_1:def 4</a></i>;<br/><a NAME="E6:45_1"/><b>then </b><i><font color="Green">E45</font></i>: 
( <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="relat_1.html#K10">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span></span>)</span> &amp; <font color="Maroon">g2</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="relat_1.html#K10">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E7:45_1"/><b>then </b>
( <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> &amp; not <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f2</font> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">g2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> &amp; not <font color="Maroon">g2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f2</font> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E8:45_1"/><b>then </b>
( not <font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> &amp; not <font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/><b>hence </b><a NAME="E9:45_1"/>
( <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> &amp; <font color="Maroon">g2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font> &amp; <font color="Maroon">x0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g2</font> &amp; <font color="Maroon">g2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> &amp; <font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:45"/><b>then </b><i><font color="Green">E46</font></i>: 
( <font color="Maroon">f2</font> <a href="rfunct_1.html#K4">^</a>  <a href="limfunc3.html#R1" target="_self">is_convergent_in</a> <font color="Maroon">x0</font> &amp;  <a href="limfunc3.html#K1" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="rfunct_1.html#K4">^</a> </span>)</span>,<font color="Maroon">x0</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="limfunc3.html#K1" target="_self">lim</a> <font color="Maroon">f2</font>,<font color="Maroon">x0</font></span>)</span> <a href="real_1.html#K2">"</a>  )
 
<b>by </b><i><a class="txt" href="limfunc3.html#E1"><i><font color="Green">Lemma31</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E5:45"/>
<font color="Maroon">f1</font> <a href="rfunct_1.html#K2">/</a> <font color="Maroon">f2</font> <a href="limfunc3.html#R1" target="_self">is_convergent_in</a> <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/>

<b>thus </b> <a href="limfunc3.html#K1" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="rfunct_1.html#K2">/</a> <font color="Maroon">f2</font></span>)</span>,<font color="Maroon">x0</font> = 
<span class="p1">(<span class="default"><a href="limfunc3.html#K1" target="_self">lim</a> <font color="Maroon">f1</font>,<font color="Maroon">x0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="limfunc3.html#K1" target="_self">lim</a> <font color="Maroon">f2</font>,<font color="Maroon">x0</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="limfunc3.html#E1"><i><font color="Green">Lemma31</font></i></a>, , , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="limfunc3.html#K1" target="_self">lim</a> <font color="Maroon">f1</font>,<font color="Maroon">x0</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="limfunc3.html#K1" target="_self">lim</a> <font color="Maroon">f2</font>,<font color="Maroon">x0</font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>
;<br/>


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