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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="struct_0.html#NM10" title="STRUCT_0:NM.10">PartFunc</a> of <font color="Maroon" title="c2">RNS</font>,<font color="Maroon" title="c1">CNS</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="c2">RNS</font>;<br/>







<b>assume </b><a NAME="E1:80"/><i><font color="Green" title="E68">A1</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="ncfcont1.html#R3" title="NCFCONT1:pred.3">is_continuous_in</a> <font color="Maroon" title="c4">x0</font>
 ;<br/>

<a NAME="E2:80"/><b>then </b><i><font color="Green" title="E69">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="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c2">RNS</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="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a> &amp;  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">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="clvect_1.html#V6" title="CLVECT_1:attr.6">convergent</a> &amp; <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="clvect_1.html#K8" title="CLVECT_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#D17" target="_self" title="NCFCONT1:def.17">Def17</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="E70">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="E69">A2</font></i></a>, <a class="ref" href="ncfcont1.html#D4" target="_self" title="NCFCONT1:def.4">Def4</a></i>;<br/><b>let </b><font color="Maroon" title="c5">s1</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c2">RNS</font>;<br/><b>assume </b><a NAME="E2:80_1"/><i><font color="Green" title="E71">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="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a> &amp;  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">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="E72">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#D4" target="_self" title="NCFCONT1:def.4">Def4</a></i>;<br/><a NAME="E4:80_1"/><b>then </b><i><font color="Green" title="E73">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="clvect_1.html#V6" title="CLVECT_1:attr.6">convergent</a> &amp; <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="clvect_1.html#K8" title="CLVECT_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="E68">A1</font></i></a>, <a class="txt" href="ncfcont1.html#E2:80_1"><i><font color="Green" title="E71">A4</font></i></a>, <a class="ref" href="ncfcont1.html#D17" target="_self" title="NCFCONT1:def.17">Def17</a></i>;<br/><a NAME="E5:80_1"/><b>then </b>
<span class="p1"><a href="clvect_1.html#K7" title="CLVECT_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="clvect_1.html#K7" title="CLVECT_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="clvect_1.html#T119" title="CLVECT_1:th.119">CLVECT_1:119</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="nfcont_1.html#K3" title="NFCONT_1:func.3">*</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="E72">A5</font></i></a>, <a class="ref" href="ncfcont1.html#T52" target="_self" title="NCFCONT1:th.52">Th52</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="partfun1.html#K7" title="PARTFUN1:func.7">/.</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="E70">A3</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span>
<b>by </b><i><a class="txt" href="ncfcont1.html#E1:80_1"><i><font color="Green" title="E70">A3</font></i></a>, <a class="ref" href="ncfcont1.html#D4" target="_self" title="NCFCONT1:def.4">Def4</a></i>
<br/>.= 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1"><a href="clvect_1.html#K7" title="CLVECT_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="clvect_1.html#K7" title="CLVECT_1:func.7">.||</a></span>
<b>by </b><i><a class="txt" href="ncfcont1.html#E4:80_1"><i><font color="Green" title="E73">A6</font></i></a>, <a class="ref" href="clopban1.html#T45" title="CLOPBAN1:th.45">CLOPBAN1:45</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="nfcont_1.html#K3" title="NFCONT_1:func.3">*</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="E72">A5</font></i></a>, <a class="ref" href="ncfcont1.html#T52" target="_self" title="NCFCONT1:th.52">Th52</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="nfcont_1.html#R2" title="NFCONT_1:pred.2">is_continuous_in</a> <font color="Maroon" title="c4">x0</font>
 <b>by </b><i><a class="ref" href="nfcont_1.html#D10" title="NFCONT_1:def.10">NFCONT_1:def 10</a></i>;<br/>

<a NAME="E5:80"/>
 <a href="vfunct_2.html#K6" title="VFUNCT_2:func.6">-</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> </span>)</span> <a href="vfunct_2.html#K4" title="VFUNCT_2:func.4">(#)</a> <font color="Maroon" title="c3">f</font>
 
<b>by </b><i><a class="ref" href="vfunct_2.html#T23" title="VFUNCT_2:th.23">VFUNCT_2:23</a></i>;<br/>
<b>hence </b><a NAME="E6:80"/>
 <a href="vfunct_2.html#K6" title="VFUNCT_2:func.6">-</a> <font color="Maroon" title="c3">f</font> <a href="ncfcont1.html#R3" title="NCFCONT1:pred.3">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="E68">A1</font></i></a>, <a class="ref" href="ncfcont1.html#T58" target="_self" title="NCFCONT1:th.58">Th58</a></i>;<br/>


</div>
