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







<b>assume </b><b>that </b><br/><a NAME="E1:52"/><i><font color="Green">E43</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="E2:52"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">Y</font> is <a href="nfcont_1.html#V1" target="_self">compact</a>
 <b>and </b><br/><a NAME="E3:52"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">f</font> <a href="nfcont_1.html#R3" target="_self">is_continuous_on</a> <font color="Maroon">Y</font>
 ;<br/>

<i><font color="Green">E57</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="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<a NAME="E5:52"/>
<font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font> <a href="nfcont_1.html#R3" target="_self">is_continuous_on</a> <font color="Maroon">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:52_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><a class="txt" href="nfcont_1.html#E1:52"><i><font color="Green">E43</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="nfcont_1.html#D11" target="_self">NFCONT_1:def 11</a><br/>

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

<b>assume </b><a NAME="E2:52_2"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<a NAME="E3:52_2"/><b>then </b>
<font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font> <a href="nfcont_1.html#R1" target="_self">is_continuous_in</a> <font color="Maroon">r</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:52_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="nfcont_1.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>
<a NAME="E6:52"/><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" target="_self">compact</a>
 
<b>by </b><i><a class="ref" href="nfcont_1.html#D1" target="_self">Def1</a>, <a class="txt" href="nfcont_1.html#E1:52"><i><font color="Green">E43</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E7:52"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Y</font> is <a href="nfcont_1.html#V1" target="_self">compact</a>
 <b>by </b><i><a class="ref" href="relat_1.html#T148">RELAT_1:148</a></i>;<br/>


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