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

<span class="kw">let </span><font color="Maroon" title="c1">CNS</font> be   <a href="clvect_1.html#NM5" title="CLVECT_1:NM.5">ComplexNormSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">RNS</font> being   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a><br/>  for <font color="Olive" title="b2">f</font> being   <a href="struct_0.html#NM12" title="STRUCT_0:NM.12">PartFunc</a> of <font color="Maroon" title="c1">CNS</font>,<font color="Olive" title="b1">RNS</font>  st  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b2">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>  &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b2">f</font> is  <a href="ncfcont1.html#V1" title="NCFCONT1:attr.1">compact</a>  &amp; <font color="Olive" title="b2">f</font> <a href="ncfcont1.html#R8" title="NCFCONT1:pred.8">is_continuous_on</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b2">f</font> holds <br/> ex <font color="Olive" title="b3">x1</font>, <font color="Olive" title="b4">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">CNS</font> st <br/>( <font color="Olive" title="b3">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b2">f</font> &amp; <font color="Olive" title="b4">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b2">f</font> &amp; <span class="p1"><a href="normsp_0.html#K3" title="NORMSP_0:func.3">||.</a><span class="default"><font color="Olive" title="b2">f</font></span><a href="normsp_0.html#K3" title="NORMSP_0:func.3">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">x1</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#K2" title="RELSET_1:func.2">rng</a> <span class="p2"><a href="normsp_0.html#K3" title="NORMSP_0:func.3">||.</a><span class="default"><font color="Olive" title="b2">f</font></span><a href="normsp_0.html#K3" title="NORMSP_0:func.3">.||</a></span></span>)</span> &amp; <span class="p1"><a href="normsp_0.html#K3" title="NORMSP_0:func.3">||.</a><span class="default"><font color="Olive" title="b2">f</font></span><a href="normsp_0.html#K3" title="NORMSP_0:func.3">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b4">x2</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#K2" title="RELSET_1:func.2">rng</a> <span class="p2"><a href="normsp_0.html#K3" title="NORMSP_0:func.3">||.</a><span class="default"><font color="Olive" title="b2">f</font></span><a href="normsp_0.html#K3" title="NORMSP_0:func.3">.||</a></span></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">RNS</font> be   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <a href="struct_0.html#NM12" title="STRUCT_0:NM.12">PartFunc</a> of <font color="Maroon" title="c1">CNS</font>,<font color="Maroon" title="c2">RNS</font>  st  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">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>  &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">f</font> is  <a href="ncfcont1.html#V1" title="NCFCONT1:attr.1">compact</a>  &amp; <font color="Olive" title="b1">f</font> <a href="ncfcont1.html#R8" title="NCFCONT1:pred.8">is_continuous_on</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">f</font> holds <br/> ex <font color="Olive" title="b2">x1</font>, <font color="Olive" title="b3">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">CNS</font> st <br/>( <font color="Olive" title="b2">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">f</font> &amp; <font color="Olive" title="b3">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">f</font> &amp; <span class="p1"><a href="normsp_0.html#K3" title="NORMSP_0:func.3">||.</a><span class="default"><font color="Olive" title="b1">f</font></span><a href="normsp_0.html#K3" title="NORMSP_0:func.3">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">x1</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#K2" title="RELSET_1:func.2">rng</a> <span class="p2"><a href="normsp_0.html#K3" title="NORMSP_0:func.3">||.</a><span class="default"><font color="Olive" title="b1">f</font></span><a href="normsp_0.html#K3" title="NORMSP_0:func.3">.||</a></span></span>)</span> &amp; <span class="p1"><a href="normsp_0.html#K3" title="NORMSP_0:func.3">||.</a><span class="default"><font color="Olive" title="b1">f</font></span><a href="normsp_0.html#K3" title="NORMSP_0:func.3">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">x2</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#K2" title="RELSET_1:func.2">rng</a> <span class="p2"><a href="normsp_0.html#K3" title="NORMSP_0:func.3">||.</a><span class="default"><font color="Olive" title="b1">f</font></span><a href="normsp_0.html#K3" title="NORMSP_0:func.3">.||</a></span></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <a href="struct_0.html#NM12" title="STRUCT_0:NM.12">PartFunc</a> of <font color="Maroon" title="c1">CNS</font>,<font color="Maroon" title="c2">RNS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="relset_1.html#K1" title="RELSET_1:func.1">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>  &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> is  <a href="ncfcont1.html#V1" title="NCFCONT1:attr.1">compact</a>  &amp; <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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> implies  ex <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">CNS</font> st <br/>( <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Olive" title="b2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> &amp; <span class="p1"><a href="normsp_0.html#K3" title="NORMSP_0:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="normsp_0.html#K3" title="NORMSP_0:func.3">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x1</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#K2" title="RELSET_1:func.2">rng</a> <span class="p2"><a href="normsp_0.html#K3" title="NORMSP_0:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="normsp_0.html#K3" title="NORMSP_0:func.3">.||</a></span></span>)</span> &amp; <span class="p1"><a href="normsp_0.html#K3" title="NORMSP_0:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="normsp_0.html#K3" title="NORMSP_0:func.3">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">x2</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#K2" title="RELSET_1:func.2">rng</a> <span class="p2"><a href="normsp_0.html#K3" title="NORMSP_0:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="normsp_0.html#K3" title="NORMSP_0:func.3">.||</a></span></span>)</span> ) )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:104"/><span class="lab"><font color="Green" title="E72">A1</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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> 
 <span class="kw">and </span><br/><a NAME="E2:104"/><span class="lab"><font color="Green" title="E73">A2</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> is  <a href="ncfcont1.html#V1" title="NCFCONT1:attr.1">compact</a> 
 <span class="kw">and </span><br/><a NAME="E3:104"/><span class="lab"><font color="Green" title="E74">A3</font></span>: 
<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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">CNS</font> st <br/>( <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Olive" title="b2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> &amp; <span class="p1"><a href="normsp_0.html#K3" title="NORMSP_0:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="normsp_0.html#K3" title="NORMSP_0:func.3">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x1</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#K2" title="RELSET_1:func.2">rng</a> <span class="p2"><a href="normsp_0.html#K3" title="NORMSP_0:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="normsp_0.html#K3" title="NORMSP_0:func.3">.||</a></span></span>)</span> &amp; <span class="p1"><a href="normsp_0.html#K3" title="NORMSP_0:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="normsp_0.html#K3" title="NORMSP_0:func.3">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">x2</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#K2" title="RELSET_1:func.2">rng</a> <span class="p2"><a href="normsp_0.html#K3" title="NORMSP_0:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="normsp_0.html#K3" title="NORMSP_0:func.3">.||</a></span></span>)</span> )</span><br/>

<a NAME="E4:104"/><span class="lab"><font color="Green" title="E75">A4</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1"><a href="normsp_0.html#K3" title="NORMSP_0:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="normsp_0.html#K3" title="NORMSP_0:func.3">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_0.html#D3" title="NORMSP_0:def.3">NORMSP_0:def 3</a></span>;<br/>
<a NAME="E5:104"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1"><a href="normsp_0.html#K3" title="NORMSP_0:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="normsp_0.html#K3" title="NORMSP_0:func.3">.||</a></span> is  <a href="ncfcont1.html#V1" title="NCFCONT1:attr.1">compact</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:104"><span class="lab"><font color="Green" title="E73">A2</font></span></a>, <a class="ref" href="normsp_0.html#D3" title="NORMSP_0:def.3">NORMSP_0:def 3</a></span>;<br/>
<a NAME="E6:104"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A5</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1"><a href="normsp_0.html#K3" title="NORMSP_0:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="normsp_0.html#K3" title="NORMSP_0:func.3">.||</a></span> is  <a href="rcomp_1.html#V1" title="RCOMP_1:attr.1">compact</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:104"><span class="lab"><font color="Green" title="E74">A3</font></span></a>, <a class="txt" href="#E4:104"><span class="lab"><font color="Green" title="E75">A4</font></span></a>, <a class="ref" href="ncfcont1.html#T72" target="_self" title="NCFCONT1:th.72">Th93</a>, <a class="ref" href="ncfcont1.html#T81" target="_self" title="NCFCONT1:th.81">Th102</a></span>;<br/>
<a NAME="E7:104"/><span class="lab"><font color="Green" title="E77">A6</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1"><a href="normsp_0.html#K3" title="NORMSP_0:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="normsp_0.html#K3" title="NORMSP_0:func.3">.||</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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:104"><span class="lab"><font color="Green" title="E72">A1</font></span></a>, <a class="txt" href="#E4:104"><span class="lab"><font color="Green" title="E75">A4</font></span></a>, <a class="ref" href="relat_1.html#T42" title="RELAT_1:th.42">RELAT_1:42</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E9:104"/><span class="lab"><font color="Green" title="E78">A7</font></span>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1"><a href="normsp_0.html#K3" title="NORMSP_0:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="normsp_0.html#K3" title="NORMSP_0:func.3">.||</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#K2" title="RELSET_1:func.2">rng</a> <span class="p2"><a href="normsp_0.html#K3" title="NORMSP_0:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="normsp_0.html#K3" title="NORMSP_0:func.3">.||</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="normsp_0.html#K3" title="NORMSP_0:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="normsp_0.html#K3" title="NORMSP_0:func.3">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:104"><span class="lab"><font color="Green" title="E76">A5</font></span></a>, <a class="ref" href="partfun1.html#T3" title="PARTFUN1:th.3">PARTFUN1:3</a>, <a class="ref" href="rcomp_1.html#T14" title="RCOMP_1:th.14">RCOMP_1:14</a></span>;<br/>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E11:104"/><span class="lab"><font color="Green" title="E79">A8</font></span>: 
( <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1"><a href="normsp_0.html#K3" title="NORMSP_0:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="normsp_0.html#K3" title="NORMSP_0:func.3">.||</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#K2" title="RELSET_1:func.2">rng</a> <span class="p2"><a href="normsp_0.html#K3" title="NORMSP_0:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="normsp_0.html#K3" title="NORMSP_0:func.3">.||</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="normsp_0.html#K3" title="NORMSP_0:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="normsp_0.html#K3" title="NORMSP_0:func.3">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:104"><span class="lab"><font color="Green" title="E77">A6</font></span></a>, <a class="txt" href="#E6:104"><span class="lab"><font color="Green" title="E76">A5</font></span></a>, <a class="ref" href="partfun1.html#T3" title="PARTFUN1:th.3">PARTFUN1:3</a>, <a class="ref" href="rcomp_1.html#T14" title="RCOMP_1:th.14">RCOMP_1:14</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c4">x</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">CNS</font> st <br/>( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Olive" title="b1">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> &amp; <span class="p1"><a href="normsp_0.html#K3" title="NORMSP_0:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="normsp_0.html#K3" title="NORMSP_0:func.3">.||</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#K2" title="RELSET_1:func.2">rng</a> <span class="p2"><a href="normsp_0.html#K3" title="NORMSP_0:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="normsp_0.html#K3" title="NORMSP_0:func.3">.||</a></span></span>)</span> &amp; <span class="p1"><a href="normsp_0.html#K3" title="NORMSP_0:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="normsp_0.html#K3" title="NORMSP_0:func.3">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x2</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#K2" title="RELSET_1:func.2">rng</a> <span class="p2"><a href="normsp_0.html#K3" title="NORMSP_0:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="normsp_0.html#K3" title="NORMSP_0:func.3">.||</a></span></span>)</span> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c5">y</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> &amp; <span class="p1"><a href="normsp_0.html#K3" title="NORMSP_0:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="normsp_0.html#K3" title="NORMSP_0:func.3">.||</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#K2" title="RELSET_1:func.2">rng</a> <span class="p2"><a href="normsp_0.html#K3" title="NORMSP_0:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="normsp_0.html#K3" title="NORMSP_0:func.3">.||</a></span></span>)</span> &amp; <span class="p1"><a href="normsp_0.html#K3" title="NORMSP_0:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="normsp_0.html#K3" title="NORMSP_0:func.3">.||</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#K2" title="RELSET_1:func.2">rng</a> <span class="p2"><a href="normsp_0.html#K3" title="NORMSP_0:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="normsp_0.html#K3" title="NORMSP_0:func.3">.||</a></span></span>)</span> )</span><br/>

<span class="kw">thus </span><a NAME="E12:104"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> &amp; <span class="p1"><a href="normsp_0.html#K3" title="NORMSP_0:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="normsp_0.html#K3" title="NORMSP_0:func.3">.||</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#K2" title="RELSET_1:func.2">rng</a> <span class="p2"><a href="normsp_0.html#K3" title="NORMSP_0:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="normsp_0.html#K3" title="NORMSP_0:func.3">.||</a></span></span>)</span> &amp; <span class="p1"><a href="normsp_0.html#K3" title="NORMSP_0:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="normsp_0.html#K3" title="NORMSP_0:func.3">.||</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#K2" title="RELSET_1:func.2">rng</a> <span class="p2"><a href="normsp_0.html#K3" title="NORMSP_0:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="normsp_0.html#K3" title="NORMSP_0:func.3">.||</a></span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:104"><span class="lab"><font color="Green" title="E78">A7</font></span></a>, <a class="txt" href="#E11:104"><span class="lab"><font color="Green" title="E79">A8</font></span></a>, <a class="ref" href="normsp_0.html#D3" title="NORMSP_0:def.3">NORMSP_0:def 3</a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
