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

<b>let </b><font color="Maroon" title="c1">CNS</font> be   <a href="clvect_1.html#NM5" title="CLVECT_1:NM.5">ComplexNormSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">RNS</font> be   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="struct_0.html#NM10" title="STRUCT_0:NM.10">PartFunc</a> of <font color="Maroon" title="c1">CNS</font>,<font color="Maroon" title="c2">RNS</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:133"/><i><font color="Green" title="E100">A1</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>and </b><br/><a NAME="E2:133"/><i><font color="Green" title="E101">A2</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> is <a href="ncfcont1.html#V1" title="NCFCONT1:attr.1">compact</a>
 <b>and </b><br/><a NAME="E3:133"/><i><font color="Green" title="E102">A3</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="ncfcont1.html#R8" title="NCFCONT1:pred.8">is_continuous_on</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 ;<br/>

<a NAME="E4:133"/><i><font color="Green" title="E103">A4</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> is <a href="ncfcont1.html#V1" title="NCFCONT1:attr.1">compact</a>
 
<b>by </b><i><a class="txt" href="ncfcont1.html#E2:133"><i><font color="Green" title="E101">A2</font></i></a>, <a class="ref" href="ncfcont1.html#D3" target="_self" title="NCFCONT1:def.3">Def3</a></i>;<br/>
<a NAME="E5:133"/><i><font color="Green" title="E104">A5</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span>
 
<b>by </b><i><a class="ref" href="ncfcont1.html#D3" target="_self" title="NCFCONT1:def.3">Def3</a></i>;<br/>
<a NAME="E6:133"/><b>then </b><i><font color="Green" title="E105">A6</font></i>: 
<span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="ncfcont1.html#R11" title="NCFCONT1:pred.11">is_continuous_on</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span>
 
<b>by </b><i><a class="txt" href="ncfcont1.html#E3:133"><i><font color="Green" title="E102">A3</font></i></a>, <a class="ref" href="ncfcont1.html#T93" target="_self" title="NCFCONT1:th.93">Th93</a></i>;<br/>
<a NAME="E7:133"/><i><font color="Green" title="E106">A7</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="ncfcont1.html#E1:133"><i><font color="Green" title="E100">A1</font></i></a>, <a class="txt" href="ncfcont1.html#E5:133"><i><font color="Green" title="E104">A5</font></i></a>, <a class="ref" href="relat_1.html#T65" title="RELAT_1:th.65">RELAT_1:65</a></i>;<br/>
<a NAME="E8:133"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> is <a href="rcomp_1.html#V1" title="RCOMP_1:attr.1">compact</a>
 
<b>by </b><i><a class="txt" href="ncfcont1.html#E4:133"><i><font color="Green" title="E103">A4</font></i></a>, <a class="txt" href="ncfcont1.html#E6:133"><i><font color="Green" title="E105">A6</font></i></a>, <a class="ref" href="ncfcont1.html#T102" target="_self" title="NCFCONT1:th.102">Th102</a></i>;<br/>
<a NAME="E9:133"/><b>then </b><i><font color="Green" title="E107">A8</font></i>: 
(  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> &amp;  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> )
 
<b>by </b><i><a class="txt" href="ncfcont1.html#E7:133"><i><font color="Green" title="E106">A7</font></i></a>, <a class="ref" href="rcomp_1.html#T32" title="RCOMP_1:th.32">RCOMP_1:32</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">CNS</font><b> such that </b><br/><a NAME="E11:133"/><i><font color="Green" title="E108">A9</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> &amp;  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26" title="PARTFUN1:th.26">PARTFUN1:26</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">CNS</font><b> such that </b><br/><a NAME="E13:133"/><i><font color="Green" title="E109">A10</font></i>: 
( <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> &amp;  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">y</font> )
 <b>by </b><i><a class="txt" href="ncfcont1.html#E9:133"><i><font color="Green" title="E107">A8</font></i></a>, <a class="ref" href="partfun1.html#T26" title="PARTFUN1:th.26">PARTFUN1:26</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">x</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c5">y</font>
;<br/>

<b>thus </b><a NAME="E14:133"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> &amp; <span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span></span>)</span> &amp; <span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span></span>)</span> )
 <b>by </b><i><a class="txt" href="ncfcont1.html#E11:133"><i><font color="Green" title="E108">A9</font></i></a>, <a class="txt" href="ncfcont1.html#E13:133"><i><font color="Green" title="E109">A10</font></i></a>, <a class="ref" href="ncfcont1.html#D3" target="_self" title="NCFCONT1:def.3">Def3</a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>


</div>
