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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>assume </b><b>that </b><br/><a NAME="E1:54"/><i><font color="Green">E43</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E2:54"/><i><font color="Green">E44</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> is <a href="nfcont_1.html#V1" target="_self">compact</a>
 <b>and </b><br/><a NAME="E3:54"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">f</font> <a href="nfcont_1.html#R3" target="_self">is_continuous_on</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E4:54"/><i><font color="Green">E57</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="nfcont_1.html#K2" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="nfcont_1.html#K2" target="_self">.||</a></span> is <a href="nfcont_1.html#V1" target="_self">compact</a>
 
<b>by </b><i><a class="txt" href="nfcont_1.html#E1:54"><i><font color="Green">E43</font></i></a>, </i>;<br/>
<a NAME="E5:54"/><i><font color="Green">E78</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="nfcont_1.html#K2" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="nfcont_1.html#K2" target="_self">.||</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E6:54"/><b>then </b><i><font color="Green">E79</font></i>: 
<span class="p1"><a href="nfcont_1.html#K2" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="nfcont_1.html#K2" target="_self">.||</a></span> <a href="nfcont_1.html#R4" target="_self">is_continuous_on</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="nfcont_1.html#K2" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="nfcont_1.html#K2" target="_self">.||</a></span>
 
<b>by </b><i><a class="txt" href="nfcont_1.html#E2:54"><i><font color="Green">E44</font></i></a>, </i>;<br/>
<a NAME="E7:54"/><i><font color="Green">E81</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1"><a href="nfcont_1.html#K2" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="nfcont_1.html#K2" target="_self">.||</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, , <a class="ref" href="relat_1.html#T65">RELAT_1:65</a></i>;<br/>
<a NAME="E8:54"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1"><a href="nfcont_1.html#K2" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="nfcont_1.html#K2" target="_self">.||</a></span> is <a href="rcomp_1.html#V1">compact</a>
 
<b>by </b><i>, <a class="ref" href="nfcont_1.html#T2" target="_self">Th2</a>, </i>;<br/>
<a NAME="E9:54"/><b>then </b><i><font color="Green">E82</font></i>: 
(  <a href="seq_4.html#K4">upper_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2"><a href="nfcont_1.html#K2" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="nfcont_1.html#K2" target="_self">.||</a></span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1"><a href="nfcont_1.html#K2" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="nfcont_1.html#K2" target="_self">.||</a></span> &amp;  <a href="seq_4.html#K5">lower_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2"><a href="nfcont_1.html#K2" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="nfcont_1.html#K2" target="_self">.||</a></span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1"><a href="nfcont_1.html#K2" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="nfcont_1.html#K2" target="_self">.||</a></span> )
 
<b>by </b><i>, <a class="ref" href="rcomp_1.html#T32">RCOMP_1:32</a></i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E11:54"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="nfcont_1.html#K2" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="nfcont_1.html#K2" target="_self">.||</a></span> &amp;  <a href="seq_4.html#K4">upper_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2"><a href="nfcont_1.html#K2" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="nfcont_1.html#K2" target="_self">.||</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="nfcont_1.html#K2" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="nfcont_1.html#K2" target="_self">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<b>consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E13:54"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="nfcont_1.html#K2" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="nfcont_1.html#K2" target="_self">.||</a></span> &amp;  <a href="seq_4.html#K5">lower_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2"><a href="nfcont_1.html#K2" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="nfcont_1.html#K2" target="_self">.||</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="nfcont_1.html#K2" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="nfcont_1.html#K2" target="_self">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="nfcont_1.html#D2" target="_self">Def2</a>, <a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<b>take </b>
<font color="Maroon">x</font>
;<br/>

<b>take </b>
<font color="Maroon">y</font>
;<br/>

<b>thus </b><a NAME="E14:54"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <span class="p1"><a href="nfcont_1.html#K2" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="nfcont_1.html#K2" target="_self">.||</a></span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="seq_4.html#K4">upper_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2"><a href="nfcont_1.html#K2" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="nfcont_1.html#K2" target="_self">.||</a></span></span>)</span> &amp; <span class="p1"><a href="nfcont_1.html#K2" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="nfcont_1.html#K2" target="_self">.||</a></span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="seq_4.html#K5">lower_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2"><a href="nfcont_1.html#K2" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="nfcont_1.html#K2" target="_self">.||</a></span></span>)</span> )
 <b>by </b><i>, , , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>


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