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<b>let </b><font color="Maroon">CNS1</font> be   <a href="clvect_1.html#NM5">ComplexNormSpace</a>;<br/><b>let </b><font color="Maroon">CNS2</font> be   <a href="clvect_1.html#NM5">ComplexNormSpace</a>;<br/><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="nfcont_1.html#NM1">PartFunc</a> of <font color="Maroon">CNS1</font>,<font color="Maroon">CNS2</font>;<br/>







<b>thus </b><a NAME="E1:97"/>
( <font color="Maroon">f</font> <a href="ncfcont1.html#R7" target="_self">is_continuous_on</a> <font color="Maroon">X</font> implies <font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font> <a href="ncfcont1.html#R7" target="_self">is_continuous_on</a> <font color="Maroon">X</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:97_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">f</font> <a href="ncfcont1.html#R7" target="_self">is_continuous_on</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:97_1"/><b>then </b>
( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; ( for <font color="Olive">r</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CNS1</font>  st <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font> <a href="ncfcont1.html#R1" target="_self">is_continuous_in</a> <font color="Olive">r</font> ) )
 
<b>by </b><i><a class="ref" href="ncfcont1.html#D19" target="_self">Def19</a></i>;<br/>
<a NAME="E3:97_1"/><b>then </b>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<b>hence </b><a NAME="E4:97_1"/>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="ncfcont1.html#D21" target="_self">NCFCONT1:def 21</a><br/>

<b>let </b><font color="Maroon">r</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CNS1</font>;<br/>

<b>assume </b><a NAME="E5:97_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E6:97_1"/><b>then </b>
<font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font> <a href="ncfcont1.html#R1" target="_self">is_continuous_in</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="ncfcont1.html#T12" target="_self">Th12</a>, <a class="ref" href="ncfcont1.html#D19" target="_self">Def19</a></i>;<br/>
<b>hence </b><a NAME="E7:97_1"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font> <a href="ncfcont1.html#R1" target="_self">is_continuous_in</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T101">RELAT_1:101</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:97"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font> <a href="ncfcont1.html#R7" target="_self">is_continuous_on</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E3:97"/><b>then </b><i><font color="Green">E54</font></i>: 
( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> &amp; ( for <font color="Olive">r</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CNS1</font>  st <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font> <a href="ncfcont1.html#R1" target="_self">is_continuous_in</a> <font color="Olive">r</font> ) )
 
<b>by </b><i><a class="ref" href="ncfcont1.html#D19" target="_self">Def19</a></i>;<br/>
<a NAME="E4:97"/><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E5:97"/>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<b>hence </b><a NAME="E6:97"/>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="ncfcont1.html#T13" target="_self">Th13</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="ncfcont1.html#D21" target="_self">NCFCONT1:def 21</a><br/>

<b>let </b><font color="Maroon">r</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CNS1</font>;<br/>

<b>assume </b><a NAME="E7:97"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E8:97"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font> <a href="ncfcont1.html#R1" target="_self">is_continuous_in</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="ncfcont1.html#T12" target="_self">Th12</a>, <a class="ref" href="ncfcont1.html#D19" target="_self">Def19</a></i>;<br/>
<b>hence </b><a NAME="E9:97"/>
<font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font> <a href="ncfcont1.html#R1" target="_self">is_continuous_in</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T101">RELAT_1:101</a></i>;<br/>


</div>
