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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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</font> being  <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>   st <font color="Olive" title="b1">f1</font> <a href="limfunc3.html#R1" title="LIMFUNC3:pred.1">is_convergent_in</a> <font color="Maroon" title="c1">x0</font> &amp; <font color="Olive" title="b2">f2</font> <a href="limfunc3.html#R1" title="LIMFUNC3:pred.1">is_convergent_in</a> <font color="Maroon" title="c1">x0</font> &amp;  <a href="limfunc3.html#K1" title="LIMFUNC3:func.1">lim</a> <font color="Olive" title="b2">f2</font>,<font color="Maroon" title="c1">x0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive" title="b3">r1</font>, <font color="Olive" title="b4">r2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b3">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font> &amp; <font color="Maroon" title="c1">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b4">r2</font> holds <br/> ex <font color="Olive" title="b5">g1</font>, <font color="Olive" title="b6">g2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b3">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b5">g1</font> &amp; <font color="Olive" title="b5">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font> &amp; <font color="Olive" title="b5">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f1</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Olive" title="b2">f2</font></span>)</span> &amp; <font color="Olive" title="b6">g2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b4">r2</font> &amp; <font color="Maroon" title="c1">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b6">g2</font> &amp; <font color="Olive" title="b6">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f1</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Olive" title="b2">f2</font></span>)</span> ) ) holds <br/>( <font color="Olive" title="b1">f1</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Olive" title="b2">f2</font> <a href="limfunc3.html#R1" title="LIMFUNC3:pred.1">is_convergent_in</a> <font color="Maroon" title="c1">x0</font> &amp;  <a href="limfunc3.html#K1" title="LIMFUNC3:func.1">lim</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f1</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Olive" title="b2">f2</font></span>)</span>,<font color="Maroon" title="c1">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="limfunc3.html#K1" title="LIMFUNC3:func.1">lim</a> <font color="Olive" title="b1">f1</font>,<font color="Maroon" title="c1">x0</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><a href="limfunc3.html#K1" title="LIMFUNC3:func.1">lim</a> <font color="Olive" title="b2">f2</font>,<font color="Maroon" title="c1">x0</font></span>)</span> )</span><br/><b>let </b><font color="Maroon" title="c2">f1</font>, <font color="Maroon" title="c3">f2</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">f1</font> <a href="limfunc3.html#R1" title="LIMFUNC3:pred.1">is_convergent_in</a> <font color="Maroon" title="c1">x0</font> &amp; <font color="Maroon" title="c3">f2</font> <a href="limfunc3.html#R1" title="LIMFUNC3:pred.1">is_convergent_in</a> <font color="Maroon" title="c1">x0</font> &amp;  <a href="limfunc3.html#K1" title="LIMFUNC3:func.1">lim</a> <font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">x0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">r2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font> &amp; <font color="Maroon" title="c1">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">r2</font> holds <br/> ex <font color="Olive" title="b3">g1</font>, <font color="Olive" title="b4">g2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b3">g1</font> &amp; <font color="Olive" title="b3">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font> &amp; <font color="Olive" title="b3">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c3">f2</font></span>)</span> &amp; <font color="Olive" title="b4">g2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">r2</font> &amp; <font color="Maroon" title="c1">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b4">g2</font> &amp; <font color="Olive" title="b4">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c3">f2</font></span>)</span> ) ) implies ( <font color="Maroon" title="c2">f1</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c3">f2</font> <a href="limfunc3.html#R1" title="LIMFUNC3:pred.1">is_convergent_in</a> <font color="Maroon" title="c1">x0</font> &amp;  <a href="limfunc3.html#K1" title="LIMFUNC3:func.1">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c3">f2</font></span>)</span>,<font color="Maroon" title="c1">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="limfunc3.html#K1" title="LIMFUNC3:func.1">lim</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c1">x0</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><a href="limfunc3.html#K1" title="LIMFUNC3:func.1">lim</a> <font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">x0</font></span>)</span> ) )</span><br/>





<b>assume </b><a NAME="E1:45"/><i><font color="Green" title="E29">A1</font></i>: 
( <font color="Maroon" title="c2">f1</font> <a href="limfunc3.html#R1" title="LIMFUNC3:pred.1">is_convergent_in</a> <font color="Maroon" title="c1">x0</font> &amp; <font color="Maroon" title="c3">f2</font> <a href="limfunc3.html#R1" title="LIMFUNC3:pred.1">is_convergent_in</a> <font color="Maroon" title="c1">x0</font> &amp;  <a href="limfunc3.html#K1" title="LIMFUNC3:func.1">lim</a> <font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">x0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">r2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font> &amp; <font color="Maroon" title="c1">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">r2</font> holds <br/> ex <font color="Olive" title="b3">g1</font>, <font color="Olive" title="b4">g2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b3">g1</font> &amp; <font color="Olive" title="b3">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font> &amp; <font color="Olive" title="b3">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c3">f2</font></span>)</span> &amp; <font color="Olive" title="b4">g2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">r2</font> &amp; <font color="Maroon" title="c1">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b4">g2</font> &amp; <font color="Olive" title="b4">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c3">f2</font></span>)</span> ) ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">f1</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c3">f2</font> <a href="limfunc3.html#R1" title="LIMFUNC3:pred.1">is_convergent_in</a> <font color="Maroon" title="c1">x0</font> &amp;  <a href="limfunc3.html#K1" title="LIMFUNC3:func.1">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c3">f2</font></span>)</span>,<font color="Maroon" title="c1">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="limfunc3.html#K1" title="LIMFUNC3:func.1">lim</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c1">x0</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><a href="limfunc3.html#K1" title="LIMFUNC3:func.1">lim</a> <font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">x0</font></span>)</span> )</span><br/>

<a NAME="E2:45"/><i><font color="Green" title="E30">A2</font></i>: 
<font color="Maroon" title="c2">f1</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c3">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#T47" title="RFUNCT_1:th.47">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" title="c4">r1</font>, <font color="Maroon" title="c5">r2</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font> &amp; <font color="Maroon" title="c1">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">r2</font> implies  ex <font color="Olive" title="b1">g1</font>, <font color="Olive" title="b2">g2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c4">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">g1</font> &amp; <font color="Olive" title="b1">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font> &amp; <font color="Olive" title="b1">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f2</font> &amp; <font color="Olive" title="b2">g2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">r2</font> &amp; <font color="Maroon" title="c1">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">g2</font> &amp; <font color="Olive" title="b2">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f2</font> &amp; <font color="Maroon" title="c3">f2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">g1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">f2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">g2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )</span><br/><b>assume </b><a NAME="E1:45_1"/>
( <font color="Maroon" title="c4">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font> &amp; <font color="Maroon" title="c1">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">r2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">g1</font>, <font color="Olive" title="b2">g2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c4">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">g1</font> &amp; <font color="Olive" title="b1">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font> &amp; <font color="Olive" title="b1">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f2</font> &amp; <font color="Olive" title="b2">g2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">r2</font> &amp; <font color="Maroon" title="c1">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">g2</font> &amp; <font color="Olive" title="b2">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f2</font> &amp; <font color="Maroon" title="c3">f2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">g1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">f2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">g2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/><b>then consider </b><font color="Maroon" title="c6">g1</font>, <font color="Maroon" title="c7">g2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:45_1"/><i><font color="Green" title="E31">A3</font></i>: 
( <font color="Maroon" title="c4">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font> &amp; <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c3">f2</font></span>)</span> &amp; <font color="Maroon" title="c7">g2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">r2</font> &amp; <font color="Maroon" title="c1">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c7">g2</font> &amp; <font color="Maroon" title="c7">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c3">f2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="limfunc3.html#E1:45"><i><font color="Green" title="E29">A1</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c8">g1</font> = <font color="Maroon" title="c6">g1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">g2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c4">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">g1</font> &amp; <font color="Maroon" title="c8">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font> &amp; <font color="Maroon" title="c8">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f2</font> &amp; <font color="Olive" title="b1">g2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">r2</font> &amp; <font color="Maroon" title="c1">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">g2</font> &amp; <font color="Olive" title="b1">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f2</font> &amp; <font color="Maroon" title="c3">f2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">g1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">f2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">g2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/><b>take </b><font color="Maroon" title="c9">g2</font> = <font color="Maroon" title="c7">g2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">g1</font> &amp; <font color="Maroon" title="c8">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font> &amp; <font color="Maroon" title="c8">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f2</font> &amp; <font color="Maroon" title="c9">g2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">r2</font> &amp; <font color="Maroon" title="c1">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c9">g2</font> &amp; <font color="Maroon" title="c9">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f2</font> &amp; <font color="Maroon" title="c3">f2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">g1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">f2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">g2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/><b>thus </b><a NAME="E4:45_1"/>
( <font color="Maroon" title="c4">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">g1</font> &amp; <font color="Maroon" title="c8">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font> )
 <b>by </b><i><a class="txt" href="limfunc3.html#E3:45_1"><i><font color="Green" title="E31">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f2</font> &amp; <font color="Maroon" title="c9">g2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">r2</font> &amp; <font color="Maroon" title="c1">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c9">g2</font> &amp; <font color="Maroon" title="c9">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f2</font> &amp; <font color="Maroon" title="c3">f2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">g1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">f2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">g2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/><a NAME="E5:45_1"/><i><font color="Green" title="E32">A4</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rfunct_1.html#D4" title="RFUNCT_1:def.4">RFUNCT_1:def 4</a></i>;<br/><a NAME="E6:45_1"/><b>then </b>
( <font color="Maroon" title="c8">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> &amp; <font color="Maroon" title="c9">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> )
 <b>by </b><i><a class="txt" href="limfunc3.html#E3:45_1"><i><font color="Green" title="E31">A3</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E7:45_1"/><b>then </b>
( <font color="Maroon" title="c8">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f2</font> &amp; not <font color="Maroon" title="c8">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f2</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c9">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f2</font> &amp; not <font color="Maroon" title="c9">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f2</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E8:45_1"/><b>then </b>
( not <font color="Maroon" title="c3">f2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; not <font color="Maroon" title="c3">f2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/><b>hence </b><a NAME="E9:45_1"/>
( <font color="Maroon" title="c8">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f2</font> &amp; <font color="Maroon" title="c9">g2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">r2</font> &amp; <font color="Maroon" title="c1">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c9">g2</font> &amp; <font color="Maroon" title="c9">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f2</font> &amp; <font color="Maroon" title="c3">f2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">g1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">f2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">g2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="limfunc3.html#E3:45_1"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="limfunc3.html#E5:45_1"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E4:45"/><b>then </b><i><font color="Green" title="E31">A5</font></i>: 
( <font color="Maroon" title="c3">f2</font> <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a>  <a href="limfunc3.html#R1" title="LIMFUNC3:pred.1">is_convergent_in</a> <font color="Maroon" title="c1">x0</font> &amp;  <a href="limfunc3.html#K1" title="LIMFUNC3:func.1">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a> </span>)</span>,<font color="Maroon" title="c1">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="limfunc3.html#K1" title="LIMFUNC3:func.1">lim</a> <font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">x0</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a>  )
 
<b>by </b><i><a class="txt" href="limfunc3.html#E1:45"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="limfunc3.html#T41" target="_self" title="LIMFUNC3:th.41">Th41</a></i>;<br/>
<b>hence </b><a NAME="E5:45"/>
<font color="Maroon" title="c2">f1</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c3">f2</font> <a href="limfunc3.html#R1" title="LIMFUNC3:pred.1">is_convergent_in</a> <font color="Maroon" title="c1">x0</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E1:45"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="limfunc3.html#E2:45"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="limfunc3.html#T42" target="_self" title="LIMFUNC3:th.42">Th42</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="limfunc3.html#K1" title="LIMFUNC3:func.1">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c3">f2</font></span>)</span>,<font color="Maroon" title="c1">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="limfunc3.html#K1" title="LIMFUNC3:func.1">lim</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c1">x0</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><a href="limfunc3.html#K1" title="LIMFUNC3:func.1">lim</a> <font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">x0</font></span>)</span></span><br/>

<b>thus </b> <a href="limfunc3.html#K1" title="LIMFUNC3:func.1">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c3">f2</font></span>)</span>,<font color="Maroon" title="c1">x0</font> = 
<span class="p1">(<span class="default"><a href="limfunc3.html#K1" title="LIMFUNC3:func.1">lim</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c1">x0</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="limfunc3.html#K1" title="LIMFUNC3:func.1">lim</a> <font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">x0</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="limfunc3.html#E1:45"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="limfunc3.html#E2:45"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="limfunc3.html#E4:45"><i><font color="Green" title="E31">A5</font></i></a>, <a class="ref" href="limfunc3.html#T42" target="_self" title="LIMFUNC3:th.42">Th42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="limfunc3.html#K1" title="LIMFUNC3:func.1">lim</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c1">x0</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><a href="limfunc3.html#K1" title="LIMFUNC3:func.1">lim</a> <font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">x0</font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9" title="XCMPLX_0:def.9">XCMPLX_0:def 9</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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