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<b>let </b><font color="Maroon" title="c1">x0</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</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="E1:9"/><i><font color="Green" title="E7">A1</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="fcont_1.html#R1" title="FCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c1">x0</font>
 ;<br/>

<a NAME="E2:9"/><b>then </b><i><font color="Green" title="E8">A2</font></i>: 
( <font color="Maroon" title="c1">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font> &amp; ( for <font color="Olive" title="b1">s1</font> being  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Olive" title="b1">s1</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="Olive" title="b1">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x0</font> holds <br/>( <font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Olive" title="b1">s1</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp; <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="Olive" title="b1">s1</font></span>)</span> ) ) )
 
<b>by </b><i><a class="ref" href="fcont_1.html#D1" target="_self" title="FCONT_1:def.1">Def1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:9_1"/><i><font color="Green" title="E9">A3</font></i>: 
<font color="Maroon" title="c1">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">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="fcont_1.html#E2:9"><i><font color="Green" title="E8">A2</font></i></a>, <a class="ref" href="seq_1.html#D10" title="SEQ_1:def.10">SEQ_1:def 10</a></i>;<br/><b>let </b><font color="Maroon" title="c3">s1</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/><b>assume </b><a NAME="E2:9_1"/><i><font color="Green" title="E10">A4</font></i>: 
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">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="c3">s1</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">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x0</font> )
 ;<br/><a NAME="E3:9_1"/><b>then </b><i><font color="Green" title="E11">A5</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font>
 <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="E4:9_1"/><b>then </b><i><font color="Green" title="E12">A6</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">s1</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp; <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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">s1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="fcont_1.html#E1:9"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="fcont_1.html#E2:9_1"><i><font color="Green" title="E10">A4</font></i></a>, <a class="ref" href="fcont_1.html#D1" target="_self" title="FCONT_1:def.1">Def1</a></i>;<br/><a NAME="E5:9_1"/><b>then </b>
 <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">s1</font></span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="ref" href="seq_4.html#T26" title="SEQ_4:th.26">SEQ_4:26</a></i>;<br/><b>hence </b><a NAME="E6:9_1"/>
<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">s1</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="fcont_1.html#E3:9_1"><i><font color="Green" title="E11">A5</font></i></a>, <a class="ref" href="rfunct_2.html#T25" title="RFUNCT_2:th.25">RFUNCT_2:25</a></i>;<br/><b>thus </b><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="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">x0</font> = 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">x0</font></span>)</span>
<b>by </b><i><a class="txt" href="fcont_1.html#E1:9_1"><i><font color="Green" title="E9">A3</font></i></a>, <a class="ref" href="seq_1.html#D10" title="SEQ_1:def.10">SEQ_1:def 10</a></i>
<br/>.= 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="seq_1.html#K22" title="SEQ_1:func.22">abs</a> <span class="p2">(<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">s1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fcont_1.html#E4:9_1"><i><font color="Green" title="E12">A6</font></i></a>, <a class="ref" href="seq_4.html#T27" title="SEQ_4:th.27">SEQ_4:27</a></i>
<br/>.= 
 <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">s1</font></span>)</span>
<b>by </b><i><a class="txt" href="fcont_1.html#E3:9_1"><i><font color="Green" title="E11">A5</font></i></a>, <a class="ref" href="rfunct_2.html#T25" title="RFUNCT_2:th.25">RFUNCT_2:25</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:9"/>
 <a href="seq_1.html#K21" title="SEQ_1:func.21">abs</a> <font color="Maroon" title="c2">f</font> <a href="fcont_1.html#R1" title="FCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c1">x0</font>
 <b>by </b><i><a class="ref" href="fcont_1.html#D1" target="_self" title="FCONT_1:def.1">Def1</a></i>;<br/>

<a NAME="E5:9"/>
 <a href="seq_1.html#K16" title="SEQ_1:func.16">-</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#T38" title="RFUNCT_1:th.38">RFUNCT_1:38</a></i>;<br/>
<b>hence </b><a NAME="E6:9"/>
 <a href="seq_1.html#K16" title="SEQ_1:func.16">-</a> <font color="Maroon" title="c2">f</font> <a href="fcont_1.html#R1" title="FCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c1">x0</font>
 <b>by </b><i><a class="txt" href="fcont_1.html#E1:9"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="fcont_1.html#T8" target="_self" title="FCONT_1:th.8">Th8</a></i>;<br/>


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