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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="E1:42"/><i><font color="Green" title="E27">A1</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="limfunc3.html#R1" title="LIMFUNC3:pred.1">is_convergent_in</a> <font color="Maroon" title="c1">x0</font>
 ;<br/>

<div><i><font color="Green" title="E28">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">r1</font>, <font color="Maroon" title="c4">r2</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>assume </b><a NAME="E1:42_1"/>
( <font color="Maroon" title="c3">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="c4">r2</font> )
 ;<br/><b>then consider </b><font color="Maroon" title="c5">g1</font>, <font color="Maroon" title="c6">g2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:42_1"/><i><font color="Green" title="E29">A3</font></i>: 
( <font color="Maroon" title="c3">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">g1</font> &amp; <font color="Maroon" title="c5">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="c5">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c6">g2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">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="c6">g2</font> &amp; <font color="Maroon" title="c6">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> )
 <b>by </b><i><a class="txt" href="limfunc3.html#E1:42"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="limfunc3.html#D1" target="_self" title="LIMFUNC3:def.1">Def1</a></i>;<br/><b>take </b><font color="Maroon" title="c7">g1</font> = <font color="Maroon" title="c5">g1</font>;<br/><b>take </b><font color="Maroon" title="c8">g2</font> = <font color="Maroon" title="c6">g2</font>;<br/><b>thus </b><a NAME="E4:42_1"/>
( <font color="Maroon" title="c3">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c7">g1</font> &amp; <font color="Maroon" title="c7">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="c7">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="seq_1.html#K21" title="SEQ_1:func.21">abs</a> <font color="Maroon" title="c2">f</font></span>)</span> &amp; <font color="Maroon" title="c8">g2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">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="c8">g2</font> &amp; <font color="Maroon" title="c8">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="seq_1.html#K21" title="SEQ_1:func.21">abs</a> <font color="Maroon" title="c2">f</font></span>)</span> )
 <b>by </b><i><a class="txt" href="limfunc3.html#E3:42_1"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="seq_1.html#D10" title="SEQ_1:def.10">SEQ_1:def 10</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E29">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2"><b>now </b></a><div class="add"><a NAME="E1:42_2"/><i><font color="Green" title="E30">A5</font></i>: 
 <a href="limfunc3.html#K1" title="LIMFUNC3:func.1">lim</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc3.html#K1" title="LIMFUNC3:func.1">lim</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">x0</font>
 ;<br/><b>let </b><font color="Maroon" title="c3">seq</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/><b>assume </b><a NAME="E2:42_2"/><i><font color="Green" title="E31">A6</font></i>: 
( <font color="Maroon" title="c3">seq</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c3">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x0</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">seq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="seq_1.html#K21" title="SEQ_1:func.21">abs</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ;<br/><a NAME="E3:42_2"/><b>then </b><i><font color="Green" title="E32">A7</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">seq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="seq_1.html#D10" title="SEQ_1:def.10">SEQ_1:def 10</a></i>;<br/><a NAME="E4:42_2"/><b>then </b><i><font color="Green" title="E33">A8</font></i>: 
( <font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c3">seq</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc3.html#K1" title="LIMFUNC3:func.1">lim</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">x0</font> )
 <b>by </b><i><a class="txt" href="limfunc3.html#E1:42"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="limfunc3.html#E1:42_2"><i><font color="Green" title="E30">A5</font></i></a>, <a class="txt" href="limfunc3.html#E2:42_2"><i><font color="Green" title="E31">A6</font></i></a>, <a class="ref" href="limfunc3.html#D4" target="_self" title="LIMFUNC3:def.4">Def4</a></i>;<br/><a NAME="E5:42_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">seq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E3:42_2"><i><font color="Green" title="E32">A7</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E6:42_2"/><b>then </b><i><font color="Green" title="E34">A9</font></i>: 
 <a href="seq_1.html#K22" title="SEQ_1:func.22">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="seq_1.html#K21" title="SEQ_1:func.21">abs</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c3">seq</font>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T25" title="RFUNCT_2:th.25">RFUNCT_2:25</a></i>;<br/><b>hence </b><a NAME="E7:42_2"/>
<span class="p1">(<span class="default"><a href="seq_1.html#K21" title="SEQ_1:func.21">abs</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c3">seq</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="limfunc3.html#E4:42_2"><i><font color="Green" title="E33">A8</font></i></a>, <a class="ref" href="seq_4.html#T26" title="SEQ_4:th.26">SEQ_4:26</a></i>;<br/><b>thus </b><a NAME="E8:42_2"/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K21" title="SEQ_1:func.21">abs</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="limfunc3.html#K1" title="LIMFUNC3:func.1">lim</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="limfunc3.html#E4:42_2"><i><font color="Green" title="E33">A8</font></i></a>, <a class="txt" href="limfunc3.html#E6:42_2"><i><font color="Green" title="E34">A9</font></i></a>, <a class="ref" href="seq_4.html#T27" title="SEQ_4:th.27">SEQ_4:27</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:42"/>
 <a href="seq_1.html#K21" title="SEQ_1:func.21">abs</a> <font color="Maroon" title="c2">f</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#E2:42"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="limfunc3.html#D1" target="_self" title="LIMFUNC3:def.1">Def1</a></i>;<br/>

<b>hence </b><a NAME="E5:42"/>
 <a href="limfunc3.html#K1" title="LIMFUNC3:func.1">lim</a> <span class="p1">(<span class="default"><a href="seq_1.html#K21" title="SEQ_1:func.21">abs</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c1">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="limfunc3.html#K1" title="LIMFUNC3:func.1">lim</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="limfunc3.html#E3:42"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="limfunc3.html#D4" target="_self" title="LIMFUNC3:def.4">Def4</a></i>;<br/>


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