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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="clvect_1.html#NM5">ComplexNormSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="normsp_1.html#NM1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nfcont_1.html#NM1">PartFunc</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:141"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E2:141"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E3:141"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="ncfcont1.html#V1" target="_self">compact</a>
 <b>and </b><br/><a NAME="E4:141"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="ncfcont1.html#R8" target="_self">is_continuous_on</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<i><font color="Green">E109</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="ncfcont1.html#E2:141"><i><font color="Green">E106</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<a NAME="E6:141"/>
<font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font> <a href="ncfcont1.html#R8" target="_self">is_continuous_on</a> <font color="Maroon">c<sub>4</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="141_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:141_2"/>
<font color="Maroon">c<sub>4</sub></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">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="ncfcont1.html#E5:141"><i><font color="Green">E109</font></i></a></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">c<sub>5</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E2:141_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E3:141_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font> <a href="ncfcont1.html#R2" target="_self">is_continuous_in</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="ncfcont1.html#E4:141"><i><font color="Green">E108</font></i></a>, <a class="ref" href="ncfcont1.html#D22" target="_self">Def22</a></i>;<br/>
<b>hence </b><a NAME="E4:141_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font> <a href="ncfcont1.html#R2" target="_self">is_continuous_in</a> <font color="Maroon">c<sub>5</sub></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">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E8:141"/><i><font color="Green">E110</font></i>: 
( <font color="Maroon">c<sub>5</sub></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">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>6</sub></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">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <span class="p1"><a href="ncfcont1.html#K3" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="ncfcont1.html#K3" target="_self">.||</a></span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></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="ncfcont1.html#K3" target="_self">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="ncfcont1.html#K3" target="_self">.||</a></span></span>)</span> &amp; <span class="p1"><a href="ncfcont1.html#K3" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="ncfcont1.html#K3" target="_self">.||</a></span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>6</sub></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="ncfcont1.html#K3" target="_self">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="ncfcont1.html#K3" target="_self">.||</a></span></span>)</span> )
 <b>by </b><i><a class="txt" href="ncfcont1.html#E1:141"><i><font color="Green">E105</font></i></a>, <a class="txt" href="ncfcont1.html#E3:141"><i><font color="Green">E107</font></i></a>, <a class="txt" href="ncfcont1.html#E5:141"><i><font color="Green">E109</font></i></a>, <a class="ref" href="ncfcont1.html#T109" target="_self">Th109</a></i>;<br/>
<a NAME="E9:141"/><i><font color="Green">E111</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="ncfcont1.html#K3" target="_self">||.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="ncfcont1.html#K3" target="_self">.||</a></span>
 
<b>by </b><i><a class="ref" href="ncfcont1.html#D3" target="_self">Def3</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<b>thus </b><a NAME="E10:141"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="ncfcont1.html#E5:141"><i><font color="Green">E109</font></i></a>, <a class="txt" href="ncfcont1.html#E8:141"><i><font color="Green">E110</font></i></a></i>;<br/>

<a NAME="E11:141"/><i><font color="Green">E112</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="ncfcont1.html#K3" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="ncfcont1.html#K3" target="_self">.||</a></span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="ncfcont1.html#K3" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="ncfcont1.html#K3" target="_self">.||</a></span> )
 
<b>by </b><i><a class="txt" href="ncfcont1.html#E8:141"><i><font color="Green">E110</font></i></a>, <a class="ref" href="ncfcont1.html#D3" target="_self">Def3</a></i>;<br/>
<b>then </b><i><font color="Green">E113</font></i>: <span class="p1"><a href="ncfcont1.html#K3" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="ncfcont1.html#K3" target="_self">.||</a></span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font> = 
<span class="p1"><a href="ncfcont1.html#K3" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="ncfcont1.html#K3" target="_self">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></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">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="txt" href="ncfcont1.html#E11:141"><i><font color="Green">E112</font></i></a>, <a class="ref" href="ncfcont1.html#D3" target="_self">Def3</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="txt" href="ncfcont1.html#E8:141"><i><font color="Green">E110</font></i></a>, <a class="ref" href="partfun2.html#T32">PARTFUN2:32</a></i>
<br/>.= 
<span class="p1"><a href="ncfcont1.html#K3" target="_self">||.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="ncfcont1.html#K3" target="_self">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="ncfcont1.html#E2:141"><i><font color="Green">E106</font></i></a>, <a class="txt" href="ncfcont1.html#E5:141"><i><font color="Green">E109</font></i></a>, <a class="txt" href="ncfcont1.html#E8:141"><i><font color="Green">E110</font></i></a>, <a class="txt" href="ncfcont1.html#E9:141"><i><font color="Green">E111</font></i></a>, <a class="ref" href="ncfcont1.html#D3" target="_self">Def3</a></i>
<br/>.= 
<span class="p1"><a href="ncfcont1.html#K3" target="_self">||.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="ncfcont1.html#K3" target="_self">.||</a></span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="ncfcont1.html#E2:141"><i><font color="Green">E106</font></i></a>, <a class="txt" href="ncfcont1.html#E5:141"><i><font color="Green">E109</font></i></a>, <a class="txt" href="ncfcont1.html#E8:141"><i><font color="Green">E110</font></i></a>, <a class="txt" href="ncfcont1.html#E9:141"><i><font color="Green">E111</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/>
<i><font color="Green">E114</font></i>: <span class="p1"><a href="ncfcont1.html#K3" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="ncfcont1.html#K3" target="_self">.||</a></span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>6</sub></font> = 
<span class="p1"><a href="ncfcont1.html#K3" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="ncfcont1.html#K3" target="_self">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="ncfcont1.html#E11:141"><i><font color="Green">E112</font></i></a>, <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">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="txt" href="ncfcont1.html#E11:141"><i><font color="Green">E112</font></i></a>, <a class="ref" href="ncfcont1.html#D3" target="_self">Def3</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="txt" href="ncfcont1.html#E8:141"><i><font color="Green">E110</font></i></a>, <a class="ref" href="partfun2.html#T32">PARTFUN2:32</a></i>
<br/>.= 
<span class="p1"><a href="ncfcont1.html#K3" target="_self">||.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="ncfcont1.html#K3" target="_self">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="ncfcont1.html#E2:141"><i><font color="Green">E106</font></i></a>, <a class="txt" href="ncfcont1.html#E5:141"><i><font color="Green">E109</font></i></a>, <a class="txt" href="ncfcont1.html#E8:141"><i><font color="Green">E110</font></i></a>, <a class="txt" href="ncfcont1.html#E9:141"><i><font color="Green">E111</font></i></a>, <a class="ref" href="ncfcont1.html#D3" target="_self">Def3</a></i>
<br/>.= 
<span class="p1"><a href="ncfcont1.html#K3" target="_self">||.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="ncfcont1.html#K3" target="_self">.||</a></span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="ncfcont1.html#E2:141"><i><font color="Green">E106</font></i></a>, <a class="txt" href="ncfcont1.html#E5:141"><i><font color="Green">E109</font></i></a>, <a class="txt" href="ncfcont1.html#E8:141"><i><font color="Green">E110</font></i></a>, <a class="txt" href="ncfcont1.html#E9:141"><i><font color="Green">E111</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/>
<span class="p1"><a href="ncfcont1.html#K3" target="_self">||.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="ncfcont1.html#K3" target="_self">.||</a></span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>4</sub></font> = 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2"><a href="ncfcont1.html#K3" target="_self">||.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="ncfcont1.html#K3" target="_self">.||</a></span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></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="ncfcont1.html#K3" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="ncfcont1.html#K3" target="_self">.||</a></span>
<b>by </b><i><a class="ref" href="ncfcont1.html#T112" target="_self">Th112</a></i>
;<br/>
<b>hence </b><a NAME="E15:141"/>
( <span class="p1"><a href="ncfcont1.html#K3" target="_self">||.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="ncfcont1.html#K3" target="_self">.||</a></span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></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="ncfcont1.html#K3" target="_self">||.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="ncfcont1.html#K3" target="_self">.||</a></span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <span class="p1"><a href="ncfcont1.html#K3" target="_self">||.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="ncfcont1.html#K3" target="_self">.||</a></span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>6</sub></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="ncfcont1.html#K3" target="_self">||.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="ncfcont1.html#K3" target="_self">.||</a></span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="ncfcont1.html#E8:141"><i><font color="Green">E110</font></i></a>, <a class="txt" href="ncfcont1.html#E12:141"><i><font color="Green">E113</font></i></a>, <a class="txt" href="ncfcont1.html#E13:141"><i><font color="Green">E114</font></i></a></i>;<br/>


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