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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="funct_1.html#V3">constant</a> <a href="goboard5.html#V2">standard</a> <a href="sprect_2.html#V1">clockwise_oriented</a> <a href="goboard5.html#NM1">special_circular_sequence</a>;<br/>

<b>assume </b><a NAME="E1:53"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K29">N-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:53"/><i><font color="Green">E31</font></i>: 
 <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">c<sub>1</sub></font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="goboard9.html#D1">GOBOARD9:def 1</a></i>;<br/>
<a NAME="E3:53"/>
 <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="goboard9.html#D1">GOBOARD9:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="struct_0.html#NM2">Subset</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"><span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span><b> such that </b><br/><a NAME="E5:53"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E6:53"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="connsp_1.html#R3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>
 <b>by </b><i><a class="ref" href="connsp_1.html#D6">CONNSP_1:def 6</a></i>;<br/>
<a NAME="E7:53"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E6:53"><i><font color="Green">E33</font></i></a>, <a class="ref" href="connsp_1.html#D5">CONNSP_1:def 5</a></i>;<br/>
<a NAME="E8:53"/><b>then </b><i><font color="Green">E34</font></i>: 
 <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E5:53"><i><font color="Green">E32</font></i></a>, <a class="ref" href="connsp_1.html#T24">CONNSP_1:24</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E10:53"/><i><font color="Green">E35</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 <b>and </b><br/><a NAME="E11:53"/><i><font color="Green">E36</font></i>: 
 <a href="pscomp_1.html#K29">N-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="sprect_3.html#T45" target="_self">Th45</a></i>;<br/>
<a NAME="E12:53"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E10:53"><i><font color="Green">E35</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E13:53"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E1:53"><i><font color="Green">E30</font></i></a>, <a class="txt" href="sprect_3.html#E11:53"><i><font color="Green">E36</font></i></a>, <a class="ref" href="sprect_3.html#T46" target="_self">Th46</a></i>;<br/>
<a NAME="E14:53"/><i><font color="Green">E39</font></i>: 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="gobrd11.html#T34">GOBRD11:34</a></i>;<br/>
<a NAME="E15:53"/><b>then </b><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/>
<a NAME="E16:53"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 4
 
<b>by </b><i><a class="ref" href="goboard7.html#T36">GOBOARD7:36</a></i>;<br/>
<a NAME="E17:53"/><b>then </b><i><font color="Green">E41</font></i>: 
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E18:53"/><i><font color="Green">E42</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E10:53"><i><font color="Green">E35</font></i></a>, <a class="txt" href="sprect_3.html#E14:53"><i><font color="Green">E39</font></i></a>, <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/>
<a NAME="E19:53"/><i><font color="Green">E43</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="sprect_3.html#E10:53"><i><font color="Green">E35</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E20:53"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E14:53"><i><font color="Green">E39</font></i></a>, <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/>
<a NAME="E21:53"/><b>then </b><i><font color="Green">E44</font></i>: 
 <a href="goboard5.html#K5">left_cell</a> <font color="Maroon">c<sub>1</sub></font>,1 <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E1:53"><i><font color="Green">E30</font></i></a>, <a class="txt" href="sprect_3.html#E11:53"><i><font color="Green">E36</font></i></a>, <a class="txt" href="sprect_3.html#E13:53"><i><font color="Green">E38</font></i></a>, <a class="txt" href="sprect_3.html#E15:53"><i><font color="Green">E40</font></i></a>, <a class="txt" href="sprect_3.html#E17:53"><i><font color="Green">E41</font></i></a>, <a class="txt" href="sprect_3.html#E18:53"><i><font color="Green">E42</font></i></a>, <a class="ref" href="goboard5.html#T29">GOBOARD5:29</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p5">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p4">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p5">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p4">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p4">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E22:53"/><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p5">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p4">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p5">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3">cell</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E10:53"><i><font color="Green">E35</font></i></a>, <a class="txt" href="sprect_3.html#E19:53"><i><font color="Green">E43</font></i></a>, <a class="ref" href="goboard6.html#T35">GOBOARD6:35</a></i>;<br/>
<a NAME="E23:53"/><i><font color="Green">E46</font></i>: 
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3">cell</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E21:53"><i><font color="Green">E44</font></i></a>, <a class="ref" href="goboard9.html#D1">GOBOARD9:def 1</a></i>;<br/>
<i><font color="Green">E47</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p0">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p5">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p0">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K22">`2</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p0">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p5">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p0">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a> </span>)</span>
<b>by </b><i><a class="ref" href="topreal3.html#T7">TOPREAL3:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p0">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p5">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p0">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> 1
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
;<br/>
<a NAME="E25:53"/><i><font color="Green">E48</font></i>: 
 <a href="pscomp_1.html#K16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="sprect_1.html#T68">SPRECT_1:68</a></i>;<br/>
<i><font color="Green">E49</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K22">`2</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i><a class="txt" href="sprect_3.html#E13:53"><i><font color="Green">E38</font></i></a>, <a class="txt" href="sprect_3.html#E14:53"><i><font color="Green">E39</font></i></a>, <a class="txt" href="sprect_3.html#E19:53"><i><font color="Green">E43</font></i></a>, <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i><a class="txt" href="sprect_3.html#E10:53"><i><font color="Green">E35</font></i></a>, <a class="txt" href="sprect_3.html#E11:53"><i><font color="Green">E36</font></i></a>, <a class="txt" href="sprect_3.html#E14:53"><i><font color="Green">E39</font></i></a>, <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>
<br/>.= 
 <a href="pscomp_1.html#K16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
;<br/>
<a NAME="E27:53"/><i><font color="Green">E50</font></i>: 
 <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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> : (  not  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  or  not <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> or  not  <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  or  not <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="sprect_3.html#T54" target="_self">Th54</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p5">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p4">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p5">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p5">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i><a class="txt" href="sprect_3.html#E1:53"><i><font color="Green">E30</font></i></a>, <a class="txt" href="sprect_3.html#E11:53"><i><font color="Green">E36</font></i></a>, <a class="txt" href="sprect_3.html#E12:53"><i><font color="Green">E37</font></i></a>, <a class="ref" href="sprect_3.html#T46" target="_self">Th46</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>
<b>by </b><i><a class="txt" href="sprect_3.html#E1:53"><i><font color="Green">E30</font></i></a>, <a class="txt" href="sprect_3.html#E11:53"><i><font color="Green">E36</font></i></a>, <a class="ref" href="topreal3.html#T9">TOPREAL3:9</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="topreal3.html#T7">TOPREAL3:7</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="sprect_3.html#E1:53"><i><font color="Green">E30</font></i></a>, <a class="txt" href="sprect_3.html#E26:53"><i><font color="Green">E49</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
<br/>.= 
 <a href="pscomp_1.html#K16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>

;<br/>
<a NAME="E29:53"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p0">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p5">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p0">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E24:53"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E30:53"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p0">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p5">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p0">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="pscomp_1.html#K16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="sprect_1.html#T68">SPRECT_1:68</a></i>;<br/>
<a NAME="E31:53"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p5">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p4">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p5">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E27:53"><i><font color="Green">E50</font></i></a></i>;<br/>
<a NAME="E32:53"/><b>then </b><i><font color="Green">E51</font></i>: 
 <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#R2">meets</a>  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E22:53"><i><font color="Green">E45</font></i></a>, <a class="txt" href="sprect_3.html#E23:53"><i><font color="Green">E46</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>] means <font color="Maroon">a<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>from </b><i><a class="ref" href="domain_1.html#S7">DOMAIN_1:sch 7</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>] means <font color="Maroon">a<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="pscomp_1.html#K16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>from </b><i><a class="ref" href="domain_1.html#S7">DOMAIN_1:sch 7</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>] means <font color="Maroon">a<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>from </b><i><a class="ref" href="domain_1.html#S7">DOMAIN_1:sch 7</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>] means <font color="Maroon">a<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>from </b><i><a class="ref" href="domain_1.html#S7">DOMAIN_1:sch 7</a>();<br/></i>
<a NAME="E37:53"/><i><font color="Green">E52</font></i>: 
 <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="sprect_1.html#T66">SPRECT_1:66</a></i>;<br/>
<a NAME="E38:53"/><i><font color="Green">E53</font></i>: 
 <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="sprect_1.html#T67">SPRECT_1:67</a></i>;<br/>
<a NAME="E39:53"/><i><font color="Green">E54</font></i>: 
 <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="sprect_1.html#T69">SPRECT_1:69</a></i>;<br/>
<a NAME="E40:53"/><i><font color="Green">E55</font></i>: 
 <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:53_4"/>
 <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:53_4_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:53_4_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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> : (  not  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  or  not <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> or  not  <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  or  not <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="sprect_3.html#E25:53"><i><font color="Green">E48</font></i></a>, <a class="txt" href="sprect_3.html#E37:53"><i><font color="Green">E52</font></i></a>, <a class="txt" href="sprect_3.html#E38:53"><i><font color="Green">E53</font></i></a>, <a class="txt" href="sprect_3.html#E39:53"><i><font color="Green">E54</font></i></a>, <a class="ref" href="sprect_3.html#T54" target="_self">Th54</a></i>;<br/>
<a NAME="E3:53_4_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></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="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; (  not  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  or  not <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> or  not  <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  or  not <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> ) )
 
;<br/>
<a NAME="E4:53_4_1"/><b>then </b>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> or <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> or <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> or <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> )
 
;<br/>
<a NAME="E5:53_4_1"/><b>then </b>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>7</sub></font> or <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E6:53_4_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:53_4"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 ;<br/>

<a NAME="E3:53_4"/><b>then </b><i><font color="Green">E56</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>7</sub></font> or <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_4_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_4_2"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> or <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> or <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> or <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> )
 </span><b>by </b><i><a class="txt" href="sprect_3.html#E3:53_4"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_4_2_1"><b>suppose </b></a><a NAME="E1:53_4_2_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><div class="add"><a NAME="E2:53_4_2_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></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">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 ;<br/><b>hence </b><a NAME="E3:53_4_2_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sprect_3.html#E38:53"><i><font color="Green">E53</font></i></a>, <a class="ref" href="sprect_3.html#T57" target="_self">Th57</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_4_2_2"><b>suppose </b></a><a NAME="E1:53_4_2_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><div class="add"><a NAME="E2:53_4_2_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></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">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="pscomp_1.html#K16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 ;<br/><b>hence </b><a NAME="E3:53_4_2_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sprect_3.html#E25:53"><i><font color="Green">E48</font></i></a>, <a class="ref" href="sprect_3.html#T57" target="_self">Th57</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_4_2_3"><b>suppose </b></a><a NAME="E1:53_4_2_3"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><div class="add"><a NAME="E2:53_4_2_3"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></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">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 ;<br/><b>hence </b><a NAME="E3:53_4_2_3"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sprect_3.html#E39:53"><i><font color="Green">E54</font></i></a>, <a class="ref" href="sprect_3.html#T57" target="_self">Th57</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_4_2_4"><b>suppose </b></a><a NAME="E1:53_4_2_4"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><div class="add"><a NAME="E2:53_4_2_4"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></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">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 ;<br/><b>hence </b><a NAME="E3:53_4_2_4"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sprect_3.html#E37:53"><i><font color="Green">E52</font></i></a>, <a class="ref" href="sprect_3.html#T57" target="_self">Th57</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E41:53"/>
for <font color="Olive">b<sub>1</sub></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 <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span> <a href="euclid.html#K22">`2</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be   <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>;<br/>

<b>assume </b><a NAME="E1:53_5"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<a NAME="E2:53_5"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></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">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 
;<br/>
<b>hence </b><a NAME="E3:53_5"/>
<font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="ref" href="jordan5d.html#T40">JORDAN5D:40</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E42:53"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="goboard8.html#T23">GOBOARD8:23</a></i>;<br/>
<a NAME="E43:53"/>
for <font color="Olive">b<sub>1</sub></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 <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be   <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>;<br/>

<b>assume </b><a NAME="E1:53_6"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<a NAME="E2:53_6"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></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">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="pscomp_1.html#K16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 
;<br/>
<b>hence </b><a NAME="E3:53_6"/>
<font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="ref" href="jordan5d.html#T42">JORDAN5D:42</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E44:53"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="goboard8.html#T24">GOBOARD8:24</a></i>;<br/>
<a NAME="E45:53"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>7</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E42:53"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xboole_1.html#T70">XBOOLE_1:70</a></i>;<br/>
<a NAME="E46:53"/>
for <font color="Olive">b<sub>1</sub></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 <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be   <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>;<br/>

<b>assume </b><a NAME="E1:53_7"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<a NAME="E2:53_7"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></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">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 
;<br/>
<b>hence </b><a NAME="E3:53_7"/>
<font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="ref" href="jordan5d.html#T41">JORDAN5D:41</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E47:53"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="goboard8.html#T22">GOBOARD8:22</a></i>;<br/>
<a NAME="E48:53"/>
for <font color="Olive">b<sub>1</sub></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 <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span> <a href="euclid.html#K21">`1</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be   <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>;<br/>

<b>assume </b><a NAME="E1:53_8"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<a NAME="E2:53_8"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></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">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 
;<br/>
<b>hence </b><a NAME="E3:53_8"/>
<font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="ref" href="jordan5d.html#T39">JORDAN5D:39</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E49:53"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="goboard8.html#T21">GOBOARD8:21</a></i>;<br/>
<a NAME="E50:53"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E47:53"><i><font color="Green">E58</font></i></a>, <a class="ref" href="xboole_1.html#T70">XBOOLE_1:70</a></i>;<br/>
<a NAME="E51:53"/><b>then </b>
 <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="subset_1.html#R1">misses</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E40:53"><i><font color="Green">E55</font></i></a>, <a class="txt" href="sprect_3.html#E45:53"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xboole_1.html#T70">XBOOLE_1:70</a></i>;<br/>
<a NAME="E52:53"/><b>then </b>
 <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="subset_1.html#T43">SUBSET_1:43</a></i>;<br/>
<b>hence </b><a NAME="E53:53"/>
 <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="sprect_3.html#E2:53"><i><font color="Green">E31</font></i></a>, <a class="txt" href="sprect_3.html#E8:53"><i><font color="Green">E34</font></i></a>, <a class="txt" href="sprect_3.html#E32:53"><i><font color="Green">E51</font></i></a>, <a class="ref" href="goboard9.html#T6">GOBOARD9:6</a></i>;<br/>


</div>
