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<b>let </b><font color="Maroon">CNS</font> be   <a href="clvect_1.html#NM5">ComplexNormSpace</a>;<br/><b>let </b><font color="Maroon">RNS</font> be   <a href="normsp_1.html#NM1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon">Y</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">CNS</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="nfcont_1.html#NM1">PartFunc</a> of <font color="Maroon">CNS</font>,<font color="Maroon">RNS</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:131"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E2:131"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">Y</font> is <a href="ncfcont1.html#V1" target="_self">compact</a>
 <b>and </b><br/><a NAME="E3:131"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">f</font> <a href="ncfcont1.html#R8" target="_self">is_continuous_on</a> <font color="Maroon">Y</font>
 ;<br/>

<i><font color="Green">E56</font></i>:  <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">Y</font></span>)</span> = 
<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#K9">/\</a> <font color="Maroon">Y</font>
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>
<br/>.= 
<font color="Maroon">Y</font>
<b>by </b><i><a class="ref" href="ncfcont1.html#T24" target="_self">Th24</a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<a NAME="E5:131"/>
<font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font> <a href="ncfcont1.html#R8" target="_self">is_continuous_on</a> <font color="Maroon">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="131_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:131_2"/>
<font color="Maroon">Y</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">Y</font></span>)</span>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="ncfcont1.html#D22" target="_self">NCFCONT1:def 22</a><br/>

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

<b>assume </b><a NAME="E2:131_2"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<a NAME="E3:131_2"/><b>then </b>
<font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font> <a href="ncfcont1.html#R2" target="_self">is_continuous_in</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="ncfcont1.html#D20" target="_self">Def20</a></i>;<br/>
<b>hence </b><a NAME="E4:131_2"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font> <a href="ncfcont1.html#R2" 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>
<a NAME="E6:131"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font></span>)</span> is <a href="nfcont_1.html#V1">compact</a>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E7:131"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Y</font> is <a href="nfcont_1.html#V1">compact</a>
 <b>by </b><i><a class="ref" href="relat_1.html#T148">RELAT_1:148</a></i>;<br/>


</div>
