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<b>let </b><font color="Maroon" title="c1">CNS</font> be   <a href="clvect_1.html#NM5" title="CLVECT_1:NM.5">ComplexNormSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">RNS</font> be   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="nfcont_1.html#NM1" title="NFCONT_1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">CNS</font>,<font color="Maroon" title="c2">RNS</font>;<br/><b>let </b><font color="Maroon" title="c4">x0</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">CNS</font>;<br/>







<b>assume </b><a NAME="E1:80"/><i><font color="Green" title="E69">A1</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="ncfcont1.html#R2" title="NCFCONT1:pred.2">is_continuous_in</a> <font color="Maroon" title="c4">x0</font>
 ;<br/>

<a NAME="E2:80"/><b>then </b><i><font color="Green" title="E70">A2</font></i>: 
( <font color="Maroon" title="c4">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="c3">f</font> &amp; ( for <font color="Olive" title="b1">s1</font> being  <a href="normsp_1.html#NM2" title="NORMSP_1:NM.2">sequence</a> of <font color="Maroon" title="c1">CNS</font>  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="c3">f</font> &amp; <font color="Olive" title="b1">s1</font> is <a href="clvect_1.html#V6" title="CLVECT_1:attr.6">convergent</a> &amp;  <a href="clvect_1.html#K11" title="CLVECT_1:func.11">lim</a> <font color="Olive" title="b1">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x0</font> holds <br/>( <font color="Maroon" title="c3">f</font> <a href="ncfcont1.html#K6" title="NCFCONT1:func.6">*</a> <font color="Olive" title="b1">s1</font> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a> &amp; <font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ncfcont1.html#K6" title="NCFCONT1:func.6">*</a> <font color="Olive" title="b1">s1</font></span>)</span> ) ) )
 
<b>by </b><i><a class="ref" href="ncfcont1.html#D16" target="_self" title="NCFCONT1:def.16">Def16</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:80_1"/><i><font color="Green" title="E71">A3</font></i>: 
<font color="Maroon" title="c4">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"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span>
 <b>by </b><i><a class="txt" href="ncfcont1.html#E2:80"><i><font color="Green" title="E70">A2</font></i></a>, <a class="ref" href="ncfcont1.html#D3" target="_self" title="NCFCONT1:def.3">Def3</a></i>;<br/><b>let </b><font color="Maroon" title="c5">s1</font> be   <a href="normsp_1.html#NM2" title="NORMSP_1:NM.2">sequence</a> of <font color="Maroon" title="c1">CNS</font>;<br/><b>assume </b><a NAME="E2:80_1"/><i><font color="Green" title="E72">A4</font></i>: 
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">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"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> &amp; <font color="Maroon" title="c5">s1</font> is <a href="clvect_1.html#V6" title="CLVECT_1:attr.6">convergent</a> &amp;  <a href="clvect_1.html#K11" title="CLVECT_1:func.11">lim</a> <font color="Maroon" title="c5">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x0</font> )
 ;<br/><a NAME="E3:80_1"/><b>then </b><i><font color="Green" title="E73">A5</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">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="c3">f</font>
 <b>by </b><i><a class="ref" href="ncfcont1.html#D3" target="_self" title="NCFCONT1:def.3">Def3</a></i>;<br/><a NAME="E4:80_1"/><b>then </b><i><font color="Green" title="E74">A6</font></i>: 
( <font color="Maroon" title="c3">f</font> <a href="ncfcont1.html#K6" title="NCFCONT1:func.6">*</a> <font color="Maroon" title="c5">s1</font> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a> &amp; <font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ncfcont1.html#K6" title="NCFCONT1:func.6">*</a> <font color="Maroon" title="c5">s1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="ncfcont1.html#E1:80"><i><font color="Green" title="E69">A1</font></i></a>, <a class="txt" href="ncfcont1.html#E2:80_1"><i><font color="Green" title="E72">A4</font></i></a>, <a class="ref" href="ncfcont1.html#D16" target="_self" title="NCFCONT1:def.16">Def16</a></i>;<br/><a NAME="E5:80_1"/><b>then </b>
<span class="p1"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ncfcont1.html#K6" title="NCFCONT1:func.6">*</a> <font color="Maroon" title="c5">s1</font></span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="ref" href="normsp_1.html#T39" title="NORMSP_1:th.39">NORMSP_1:39</a></i>;<br/><b>hence </b><a NAME="E6:80_1"/>
<span class="p1"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> <a href="ncfcont1.html#K10" title="NCFCONT1:func.10">*</a> <font color="Maroon" title="c5">s1</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="ncfcont1.html#E3:80_1"><i><font color="Green" title="E73">A5</font></i></a>, <a class="ref" href="ncfcont1.html#T51" target="_self" title="NCFCONT1:th.51">Th51</a></i>;<br/><b>thus </b><span class="p1"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">x0</font> = 
<span class="p1"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x0</font>
<b>by </b><i><a class="txt" href="ncfcont1.html#E1:80_1"><i><font color="Green" title="E71">A3</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
<b>by </b><i><a class="txt" href="ncfcont1.html#E1:80_1"><i><font color="Green" title="E71">A3</font></i></a>, <a class="ref" href="ncfcont1.html#D3" target="_self" title="NCFCONT1:def.3">Def3</a></i>
<br/>.= 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ncfcont1.html#K6" title="NCFCONT1:func.6">*</a> <font color="Maroon" title="c5">s1</font></span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span>
<b>by </b><i><a class="txt" href="ncfcont1.html#E4:80_1"><i><font color="Green" title="E74">A6</font></i></a>, <a class="ref" href="lopban_1.html#T47" title="LOPBAN_1:th.47">LOPBAN_1:47</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"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> <a href="ncfcont1.html#K10" title="NCFCONT1:func.10">*</a> <font color="Maroon" title="c5">s1</font></span>)</span>
<b>by </b><i><a class="txt" href="ncfcont1.html#E3:80_1"><i><font color="Green" title="E73">A5</font></i></a>, <a class="ref" href="ncfcont1.html#T51" target="_self" title="NCFCONT1:th.51">Th51</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:80"/>
<span class="p1"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> <a href="ncfcont1.html#R5" title="NCFCONT1:pred.5">is_continuous_in</a> <font color="Maroon" title="c4">x0</font>
 <b>by </b><i><a class="ref" href="ncfcont1.html#D19" target="_self" title="NCFCONT1:def.19">Def19</a></i>;<br/>

<a NAME="E5:80"/>
 <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">-</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c3">f</font>
 
<b>by </b><i><a class="ref" href="vfunct_1.html#T29" title="VFUNCT_1:th.29">VFUNCT_1:29</a></i>;<br/>
<b>hence </b><a NAME="E6:80"/>
 <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">-</a> <font color="Maroon" title="c3">f</font> <a href="ncfcont1.html#R2" title="NCFCONT1:pred.2">is_continuous_in</a> <font color="Maroon" title="c4">x0</font>
 <b>by </b><i><a class="txt" href="ncfcont1.html#E1:80"><i><font color="Green" title="E69">A1</font></i></a>, <a class="ref" href="ncfcont1.html#T57" target="_self" title="NCFCONT1:th.57">Th57</a></i>;<br/>


</div>
