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

<b>let </b><font color="Maroon" title="c1">s1</font>, <font color="Maroon" title="c2">t1</font>, <font color="Maroon" title="c3">s2</font>, <font color="Maroon" title="c4">t2</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c5">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>









<b>assume </b><a NAME="E1:23"/>
<font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">t</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> where <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : (  not <font color="Maroon" title="c1">s1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">s</font> or  not <font color="Olive" title="b1">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">s2</font> or  not <font color="Maroon" title="c2">t1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">t</font> or  not <font color="Olive" title="b2">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">t2</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:23"/><b>then </b><i><font color="Green" title="E22">A1</font></i>: 
<font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">{<span class="default"> <span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">s3</font>,<font color="Olive" title="b2">t3</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> where <font color="Olive" title="b1">s3</font>, <font color="Olive" title="b2">t3</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Olive" title="b1">s3</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">s1</font> </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">{<span class="default"> <span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">s4</font>,<font color="Olive" title="b2">t4</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> where <font color="Olive" title="b1">s4</font>, <font color="Olive" title="b2">t4</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Olive" title="b2">t4</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">t1</font> </span>}</span> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">{<span class="default"> <span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">s5</font>,<font color="Olive" title="b2">t5</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> where <font color="Olive" title="b1">s5</font>, <font color="Olive" title="b2">t5</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Maroon" title="c3">s2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">s5</font> </span>}</span> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">s6</font>,<font color="Olive" title="b2">t6</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> where <font color="Olive" title="b1">s6</font>, <font color="Olive" title="b2">t6</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Maroon" title="c4">t2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">t6</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="jordan1.html#T13" target="_self" title="JORDAN1:th.13">Th13</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">A0</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">t</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> where <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Olive" title="b1">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">s1</font> </span>}</span> , <font color="Maroon" title="c7">A1</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">t</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> where <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Olive" title="b2">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">t1</font> </span>}</span> , <font color="Maroon" title="c8">A2</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">t</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> where <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Maroon" title="c3">s2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">s</font> </span>}</span> , <font color="Maroon" title="c9">A3</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">t</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> where <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Maroon" title="c4">t2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">t</font> </span>}</span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="jordan1.html#E12"><i><font color="Green" title="E8">Lm2</font></i></a>, <a class="txt" href="jordan1.html#E13"><i><font color="Green" title="E9">Lm3</font></i></a>, <a class="txt" href="jordan1.html#E14"><i><font color="Green" title="E10">Lm4</font></i></a>, <a class="txt" href="jordan1.html#E15"><i><font color="Green" title="E11">Lm5</font></i></a></i>;<br/>
<a NAME="E4:23"/><i><font color="Green" title="E23">A2</font></i>: 
<font color="Maroon" title="c1">s1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">s1</font>
 
<b>by </b><i><a class="ref" href="real_2.html#T174" title="REAL_2:th.174">REAL_2:174</a></i>;<br/>
<a NAME="E5:23"/><i><font color="Green" title="E24">A3</font></i>: 
<font color="Maroon" title="c2">t1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">t1</font>
 
<b>by </b><i><a class="ref" href="real_2.html#T174" title="REAL_2:th.174">REAL_2:174</a></i>;<br/>
<a NAME="E6:23"/><i><font color="Green" title="E25">A4</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">s1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">t1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A0</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E4:23"><i><font color="Green" title="E23">A2</font></i></a></i>;<br/>
<a NAME="E7:23"/>
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">s1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">t1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E5:23"><i><font color="Green" title="E24">A3</font></i></a></i>;<br/>
<a NAME="E8:23"/><b>then </b>
<font color="Maroon" title="c6">A0</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">A1</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>by </b><i><a class="txt" href="jordan1.html#E6:23"><i><font color="Green" title="E25">A4</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E9:23"/><b>then </b><i><font color="Green" title="E26">A5</font></i>: 
<font color="Maroon" title="c6">A0</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c7">A1</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E10:23"/><i><font color="Green" title="E27">A6</font></i>: 
<font color="Maroon" title="c3">s2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">s2</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 
<b>by </b><i><a class="ref" href="real_2.html#T174" title="REAL_2:th.174">REAL_2:174</a></i>;<br/>
<a NAME="E11:23"/><i><font color="Green" title="E28">A7</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">t1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E5:23"><i><font color="Green" title="E24">A3</font></i></a></i>;<br/>
<a NAME="E12:23"/>
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">t1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A2</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E10:23"><i><font color="Green" title="E27">A6</font></i></a></i>;<br/>
<a NAME="E13:23"/><b>then </b>
<font color="Maroon" title="c7">A1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">A2</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>by </b><i><a class="txt" href="jordan1.html#E11:23"><i><font color="Green" title="E28">A7</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E14:23"/><b>then </b><i><font color="Green" title="E29">A8</font></i>: 
<font color="Maroon" title="c7">A1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c8">A2</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E15:23"/><i><font color="Green" title="E30">A9</font></i>: 
<font color="Maroon" title="c4">t2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">t2</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 
<b>by </b><i><a class="ref" href="real_2.html#T174" title="REAL_2:th.174">REAL_2:174</a></i>;<br/>
<a NAME="E16:23"/><i><font color="Green" title="E31">A10</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A2</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E10:23"><i><font color="Green" title="E27">A6</font></i></a></i>;<br/>
<a NAME="E17:23"/>
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">A3</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E15:23"><i><font color="Green" title="E30">A9</font></i></a></i>;<br/>
<a NAME="E18:23"/><b>then </b>
<font color="Maroon" title="c8">A2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">A3</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>by </b><i><a class="txt" href="jordan1.html#E16:23"><i><font color="Green" title="E31">A10</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E19:23"/><b>then </b><i><font color="Green" title="E32">A11</font></i>: 
<font color="Maroon" title="c8">A2</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c9">A3</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E20:23"/><i><font color="Green" title="E33">A12</font></i>: 
<font color="Maroon" title="c6">A0</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 
<b>by </b><i><a class="ref" href="jordan1.html#T9" target="_self" title="JORDAN1:th.9">Th9</a>, <a class="ref" href="jordan1.html#T18" target="_self" title="JORDAN1:th.18">Th18</a></i>;<br/>
<a NAME="E21:23"/><i><font color="Green" title="E34">A13</font></i>: 
<font color="Maroon" title="c7">A1</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 
<b>by </b><i><a class="ref" href="jordan1.html#T9" target="_self" title="JORDAN1:th.9">Th9</a>, <a class="ref" href="jordan1.html#T22" target="_self" title="JORDAN1:th.22">Th22</a></i>;<br/>
<a NAME="E22:23"/><i><font color="Green" title="E35">A14</font></i>: 
<font color="Maroon" title="c8">A2</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 
<b>by </b><i><a class="ref" href="jordan1.html#T9" target="_self" title="JORDAN1:th.9">Th9</a>, <a class="ref" href="jordan1.html#T16" target="_self" title="JORDAN1:th.16">Th16</a></i>;<br/>
<a NAME="E23:23"/>
<font color="Maroon" title="c9">A3</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 
<b>by </b><i><a class="ref" href="jordan1.html#T9" target="_self" title="JORDAN1:th.9">Th9</a>, <a class="ref" href="jordan1.html#T20" target="_self" title="JORDAN1:th.20">Th20</a></i>;<br/>
<b>hence </b><a NAME="E24:23"/>
<font color="Maroon" title="c5">P</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 <b>by </b><i><a class="txt" href="jordan1.html#E2:23"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="jordan1.html#E9:23"><i><font color="Green" title="E26">A5</font></i></a>, <a class="txt" href="jordan1.html#E14:23"><i><font color="Green" title="E29">A8</font></i></a>, <a class="txt" href="jordan1.html#E19:23"><i><font color="Green" title="E32">A11</font></i></a>, <a class="txt" href="jordan1.html#E20:23"><i><font color="Green" title="E33">A12</font></i></a>, <a class="txt" href="jordan1.html#E21:23"><i><font color="Green" title="E34">A13</font></i></a>, <a class="txt" href="jordan1.html#E22:23"><i><font color="Green" title="E35">A14</font></i></a>, <a class="ref" href="jordan1.html#T8" target="_self" title="JORDAN1:th.8">Th8</a></i>;<br/>


</div>
