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

<b>let </b><font color="Maroon">a</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">b</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">c</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">d</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:59"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> )
 ;<br/>

<b>set </b><font color="Maroon">X</font> =  <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font>;<br/>
<b>reconsider </b><font color="Maroon">Z</font> = <span class="p1">(<span class="default"><a href="pscomp_1.html#K13">proj1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span></span>)</span> as   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  ;<br/>
<a NAME="E3:59"/><i><font color="Green">E30</font></i>: 
 <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E4:59"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">p</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> holds <br/><font color="Olive">p</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:59_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/>

<b>then consider </b><font color="Maroon">p0</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:59_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">p0</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E4:59_1"/>
<font color="Maroon">p0</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E5:59_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K13">proj1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">p0</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T115">FUNCT_2:115</a></i>;<br/>
<b>reconsider </b><font color="Maroon">p0</font> = <font color="Maroon">p0</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i>, </i>;<br/>
<a NAME="E7:59_1"/>
 <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( <font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Olive">q</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> &amp; <font color="Olive">q</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c</font> ) or ( <font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> &amp; <font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a</font> &amp; <font color="Olive">q</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> ) or ( <font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> &amp; <font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a</font> &amp; <font color="Olive">q</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> ) or ( <font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Olive">q</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> &amp; <font color="Olive">q</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c</font> ) ) </span>}</span> 
 
<b>by </b><i>, <a class="ref" href="sppol_2.html#T58">SPPOL_2:58</a></i>;<br/>
<a NAME="E8:59_1"/><b>then </b>
 ex <font color="Olive">q</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Maroon">p0</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> &amp; ( ( <font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Olive">q</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> &amp; <font color="Olive">q</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c</font> ) or ( <font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> &amp; <font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a</font> &amp; <font color="Olive">q</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> ) or ( <font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> &amp; <font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a</font> &amp; <font color="Olive">q</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> ) or ( <font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Olive">q</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> &amp; <font color="Olive">q</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c</font> ) ) )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E9:59_1"/>
<font color="Maroon">p</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a</font>
 <b>by </b><i>, , , <a class="ref" href="jgraph_6.html#T12" target="_self">Th12</a>, <a class="ref" href="pscomp_1.html#T69">PSCOMP_1:69</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:59"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">q</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st ( for <font color="Olive">p</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> holds <br/><font color="Olive">p</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">q</font> ) holds <br/><font color="Maroon">a</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:59_2"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">p</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> holds <br/><font color="Olive">p</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">q</font>
 ;<br/>

<a NAME="E2:59_2"/>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>
<a NAME="E3:59_2"/><b>then </b><i><font color="Green">E76</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E4:59_2"/>
 <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="sppol_2.html#D3">SPPOL_2:def 3</a></i>;<br/>
<a NAME="E5:59_2"/><b>then </b><i><font color="Green">E77</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><a href="pscomp_1.html#K13">proj1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span> = 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i><a class="ref" href="pscomp_1.html#T69">PSCOMP_1:69</a></i>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
;<br/>
<a NAME="E7:59_2"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 
<b>by </b><i>, , <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<b>hence </b><a NAME="E8:59_2"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T12" target="_self">Th12</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b> <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> = 
 <a href="pscomp_1.html#K10">inf</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K13">proj1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pscomp_1.html#D30">PSCOMP_1:def 30</a></i>
<br/>.= 
 <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">Z</font>
<b>by </b><i><a class="ref" href="pscomp_1.html#D20">PSCOMP_1:def 20</a></i>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="jgraph_6.html#T11" target="_self">Th11</a>, , <a class="ref" href="seq_4.html#T61">SEQ_4:61</a></i>
;<br/>


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