<?xml version="1.0"?>
<div class="add">

<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="E1:56"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E2:56"/><i><font color="Green">E44</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="E3:56"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">Y</font> is <a href="nfcont_1.html#V1" target="_self">compact</a>
 <b>and </b><br/><a NAME="E4:56"/><i><font color="Green">E57</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">E78</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="txt" href="nfcont_1.html#E2:56"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<a NAME="E6:56"/>
<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="56_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:56_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="ref" href="nfcont_1.html#T2" target="_self">Th2</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="E2:56_2"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<a NAME="E3:56_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="E4:56_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>
<b>then consider </b><font color="Maroon">x1</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>, <font color="Maroon">x2</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E8:56"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</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> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</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> &amp; <span class="p1"><a href="nfcont_1.html#K2" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font></span>)</span></span><a href="nfcont_1.html#K2" target="_self">.||</a></span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x1</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"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font></span>)</span></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"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font></span>)</span></span><a href="nfcont_1.html#K2" target="_self">.||</a></span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x2</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"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font></span>)</span></span><a href="nfcont_1.html#K2" target="_self">.||</a></span></span>)</span> )
 <b>by </b><i><a class="txt" href="nfcont_1.html#E1:56"><i><font color="Green">E43</font></i></a>, , <a class="ref" href="nfcont_1.html#T2" target="_self">Th2</a>, <a class="ref" href="nfcont_1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E9:56"/><i><font color="Green">E81</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/>
<b>take </b>
<font color="Maroon">x1</font>
;<br/>

<b>take </b>
<font color="Maroon">x2</font>
;<br/>

<b>thus </b><a NAME="E10:56"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 <b>by </b><i><a class="ref" href="nfcont_1.html#T2" target="_self">Th2</a>, </i>;<br/>

<a NAME="E11:56"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">x1</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"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font></span>)</span></span><a href="nfcont_1.html#K2" target="_self">.||</a></span> &amp; <font color="Maroon">x2</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"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font></span>)</span></span><a href="nfcont_1.html#K2" target="_self">.||</a></span> )
 
<b>by </b><i>, </i>;<br/>
<i><font color="Green">E83</font></i>: <span class="p1"><a href="nfcont_1.html#K2" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font></span>)</span></span><a href="nfcont_1.html#K2" target="_self">.||</a></span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x1</font> = 
<span class="p1"><a href="nfcont_1.html#K2" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font></span>)</span></span><a href="nfcont_1.html#K2" target="_self">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x1</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x1</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i>, <a class="ref" href="partfun2.html#T32">PARTFUN2:32</a></i>
<br/>.= 
<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">x1</font>
<b>by </b><i><a class="txt" href="nfcont_1.html#E2:56"><i><font color="Green">E44</font></i></a>, <a class="ref" href="nfcont_1.html#T2" target="_self">Th2</a>, , <a class="ref" href="nfcont_1.html#D2" target="_self">Def2</a>, </i>
<br/>.= 
<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">x1</font>
<b>by </b><i><a class="txt" href="nfcont_1.html#E2:56"><i><font color="Green">E44</font></i></a>, <a class="ref" href="nfcont_1.html#T2" target="_self">Th2</a>, , <a class="ref" href="nfcont_1.html#D2" target="_self">Def2</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/>
<i><font color="Green">E84</font></i>: <span class="p1"><a href="nfcont_1.html#K2" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font></span>)</span></span><a href="nfcont_1.html#K2" target="_self">.||</a></span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x2</font> = 
<span class="p1"><a href="nfcont_1.html#K2" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font></span>)</span></span><a href="nfcont_1.html#K2" target="_self">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x2</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x2</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i>, <a class="ref" href="partfun2.html#T32">PARTFUN2:32</a></i>
<br/>.= 
<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">x2</font>
<b>by </b><i><a class="txt" href="nfcont_1.html#E2:56"><i><font color="Green">E44</font></i></a>, <a class="ref" href="nfcont_1.html#T2" target="_self">Th2</a>, , <a class="ref" href="nfcont_1.html#D2" target="_self">Def2</a>, </i>
<br/>.= 
<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">x2</font>
<b>by </b><i><a class="txt" href="nfcont_1.html#E2:56"><i><font color="Green">E44</font></i></a>, <a class="ref" href="nfcont_1.html#T2" target="_self">Th2</a>, , <a class="ref" href="nfcont_1.html#D2" target="_self">Def2</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/>
<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="relset_1.html#K10">.:</a> <font color="Maroon">Y</font> = 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><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> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T148">RELAT_1:148</a></i>
<br/>.= 
 <a href="relset_1.html#K5">rng</a> <span class="p1"><a href="nfcont_1.html#K2" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font></span>)</span></span><a href="nfcont_1.html#K2" target="_self">.||</a></span>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E15:56"/>
( <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">x1</font> <a href="hidden.html#R1">=</a>  <a href="seq_4.html#K4">upper_bound</a> <span class="p1">(<span class="default"><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> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Y</font></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">x2</font> <a href="hidden.html#R1">=</a>  <a href="seq_4.html#K5">lower_bound</a> <span class="p1">(<span class="default"><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> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Y</font></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="nfcont_1.html#D3" target="_self">Def3</a></i>;<br/>


</div>
