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

<b>let </b><font color="Maroon">p</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>let </b><font color="Maroon">f</font> be  non <a href="funct_1.html#V3">constant</a> <a href="goboard5.html#V2">standard</a> <a href="goboard5.html#NM1">special_circular_sequence</a>;<br/><b>let </b><font color="Maroon">p1</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>let </b><font color="Maroon">p2</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><b>that </b><br/><a NAME="E1:41"/><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="tarski.html#K1">}</a></span>
 <b>and </b><br/><a NAME="E2:41"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  or <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  )
 ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E3:41"/><i><font color="Green">E45</font></i>: 
( not <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font> &amp; not <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font> )
 <b>and </b><br/><a NAME="E4:41"/><i><font color="Green">E54</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 ;<br/>

<a NAME="E5:41"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E6:41"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 
<b>by </b><i><a class="txt" href="jordan12.html#E1"><i><font color="Green">Lemma40</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E7:41"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p2</font>,<font color="Maroon">p1</font>
 
<b>by </b><i><a class="txt" href="jordan12.html#E1"><i><font color="Green">Lemma40</font></i></a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E8:41"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">f</font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="goboard5.html#D5">GOBOARD5:def 5</a></i>;<br/>
<a NAME="E9:41"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="txt" href="jordan12.html#E1"><i><font color="Green">Lemma40</font></i></a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E10:41"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</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> <font color="Maroon">f</font> ) </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">LS</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:41"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">LS</font> &amp; <font color="Maroon">LS</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</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> <font color="Maroon">f</font> ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E14:41"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">LS</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</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> <font color="Maroon">f</font> )
 <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">G</font> =  <a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font>;<br/>
<div><i><font color="Green">E61</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">r</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="E1:41_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:41_1"/>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>;<br/><b>hence </b><a NAME="E3:41_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan12.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E4:41_1"/>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="jordan12.html#E2"><i><font color="Green">Lemma47</font></i></a>, <a class="txt" href="jordan12.html#E3"><i><font color="Green">Lemma49</font></i></a>, <a class="ref" href="jordan9.html#T17">JORDAN9:17</a></i>;<br/><b>hence </b><a NAME="E5:41_1"/>
not <font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="jordan12.html#T4" target="_self">Th4</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E63</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">r</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="E1:41_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:41_2"/>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>;<br/><b>hence </b><a NAME="E3:41_2"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E4:41_2"/>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="jordan12.html#E2"><i><font color="Green">Lemma47</font></i></a>, <a class="txt" href="jordan12.html#E3"><i><font color="Green">Lemma49</font></i></a>, <a class="ref" href="jordan9.html#T17">JORDAN9:17</a></i>;<br/><b>hence </b><a NAME="E5:41_2"/>
not <font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">i1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E18:41"/><i><font color="Green">E65</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1</font>,<font color="Maroon">j1</font></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">f</font></span>)</span> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font> )
 <b>and </b><br/><a NAME="E19:41"/><i><font color="Green">E66</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<font color="Maroon">j2</font></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">f</font></span>)</span> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font> )
 <b>and </b><br/><a NAME="E20:41"/><i><font color="Green">E67</font></i>: 
( ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan12.html#E2"><i><font color="Green">Lemma47</font></i></a>, <a class="txt" href="jordan12.html#E3"><i><font color="Green">Lemma49</font></i></a>, <a class="ref" href="jordan8.html#T6">JORDAN8:6</a></i>;<br/>
<a NAME="E21:41"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="txt" href="jordan12.html#E3"><i><font color="Green">Lemma49</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E22:41"/><b>then </b><i><font color="Green">E68</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="txt" href="jordan12.html#E3"><i><font color="Green">Lemma49</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E23:41"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E24:41"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="txt" href="jordan12.html#E5:41"><i><font color="Green">E55</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E25:41"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan12.html#E3"><i><font color="Green">Lemma49</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E26:41"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="txt" href="jordan12.html#E3"><i><font color="Green">Lemma49</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E27:41"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E28:41"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="txt" href="jordan12.html#E7:41"><i><font color="Green">E57</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E29:41"/>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="topreal3.html#T26">TOPREAL3:26</a></i>;<br/>
<a NAME="E30:41"/><b>then </b><i><font color="Green">E88</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="jordan12.html#E1"><i><font color="Green">Lemma40</font></i></a>, , <a class="txt" href="jordan12.html#E3"><i><font color="Green">Lemma49</font></i></a>, <a class="ref" href="sprect_2.html#T2">SPRECT_2:2</a></i>;<br/>
<a NAME="E31:41"/>
(  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> is <a href="sppol_1.html#V2">vertical</a> or  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> is <a href="sppol_1.html#V1">horizontal</a> )
 
<b>by </b><i><a class="ref" href="sppol_1.html#T41">SPPOL_1:41</a></i>;<br/>
<a NAME="E32:41"/><b>then </b>
(  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> is <a href="sppol_1.html#V2">vertical</a> or  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> is <a href="sppol_1.html#V1">horizontal</a> )
 
<b>by </b><i><a class="txt" href="jordan12.html#E3"><i><font color="Green">Lemma49</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/>
<a NAME="E33:41"/><b>then </b><i><font color="Green">E89</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 
<b>by </b><i><a class="ref" href="sppol_1.html#T36">SPPOL_1:36</a>, <a class="ref" href="sppol_1.html#T37">SPPOL_1:37</a></i>;<br/>
<a NAME="E34:41"/><i><font color="Green">E90</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="jordan12.html#E3"><i><font color="Green">Lemma49</font></i></a>, <a class="txt" href="jordan12.html#E12:41"><i><font color="Green">E59</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/>
<a NAME="E35:41"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p2</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> )
 
<b>by </b><i><a class="txt" href="jordan12.html#E1"><i><font color="Green">Lemma40</font></i></a>, <a class="txt" href="jordan12.html#E1:41"><i><font color="Green">E42</font></i></a>, <a class="txt" href="jordan12.html#E2:41"><i><font color="Green">E43</font></i></a>, , <a class="txt" href="jordan12.html#E3:41"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan12.html#E6:41"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E36:41"/><b>then </b><i><font color="Green">E92</font></i>: 
( <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  implies <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> )
 
<b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, , <a class="txt" href="jordan12.html#E14:41"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan12.html#E15:41"><i><font color="Green">E61</font></i></a>, , <a class="ref" href="jordan1g.html#T7">JORDAN1G:7</a></i>;<br/>
<a NAME="E37:41"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  implies <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 
<b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, , <a class="txt" href="jordan12.html#E14:41"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan12.html#E15:41"><i><font color="Green">E61</font></i></a>, , , <a class="ref" href="jordan1g.html#T6">JORDAN1G:6</a></i>;<br/>
<a NAME="E38:41"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">i2</font> <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">f</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/>
<a NAME="E39:41"/><i><font color="Green">E139</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font>
 
<b>by </b><i>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/>
<a NAME="E40:41"/><i><font color="Green">E143</font></i>: 
<font color="Maroon">i1</font> <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">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/>
<a NAME="E41:41"/><i><font color="Green">E144</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font>
 
<b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/>
<a NAME="E42:41"/><i><font color="Green">E145</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j1</font>
 
<b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/>
<a NAME="E43:41"/><i><font color="Green">E146</font></i>: 
<font color="Maroon">j1</font> <a href="xxreal_0.html#R1">&lt;=</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">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/>
<a NAME="E44:41"/><i><font color="Green">E147</font></i>: 
<font color="Maroon">j2</font> <a href="xxreal_0.html#R1">&lt;=</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">f</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/>
<a NAME="E45:41"/><i><font color="Green">E148</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2</font>
 
<b>by </b><i>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/>
<a NAME="E46:41"/><i><font color="Green">E149</font></i>: 
<font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j1</font>
 
<b>by </b><i><a class="txt" href="jordan12.html#E24:41"><i><font color="Green">E69</font></i></a>, <a class="ref" href="jordan5b.html#T1">JORDAN5B:1</a></i>;<br/>
<a NAME="E47:41"/><i><font color="Green">E150</font></i>: 
<font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i1</font>
 
<b>by </b><i><a class="txt" href="jordan12.html#E22:41"><i><font color="Green">E68</font></i></a>, <a class="ref" href="jordan5b.html#T1">JORDAN5B:1</a></i>;<br/>
<a NAME="E48:41"/><i><font color="Green">E151</font></i>: 
<font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i2</font>
 
<b>by </b><i><a class="txt" href="jordan12.html#E19:41"><i><font color="Green">E66</font></i></a>, <a class="ref" href="jordan5b.html#T1">JORDAN5B:1</a></i>;<br/>
<a NAME="E49:41"/><i><font color="Green">E152</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="jordan12.html#E3"><i><font color="Green">Lemma49</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/>
<a NAME="E50:41"/><i><font color="Green">E153</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan12.html#E2:41"><i><font color="Green">E43</font></i></a>, <a class="txt" href="jordan12.html#E3:41"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan12.html#E8:41"><i><font color="Green">E58</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<div><i><font color="Green">E160</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_3"/><i><font color="Green">E161</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><a NAME="E2:41_3"/><b>then </b><i><font color="Green">E162</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</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">f</font></span>)</span> )
 <b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E3:41_3"/>
<font color="Maroon">j2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j1</font>
 <b>by </b><i><a class="txt" href="jordan12.html#E28:41"><i><font color="Green">E79</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:41_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, , <a class="txt" href="jordan12.html#E20:41"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan12.html#E22:41"><i><font color="Green">E68</font></i></a>, <a class="ref" href="jordan12.html#T6" target="_self">Th6</a>, , <a class="txt" href="jordan12.html#E28:41"><i><font color="Green">E79</font></i></a>, <a class="ref" href="goboard5.html#T5">GOBOARD5:5</a></i>;<br/><a NAME="E5:41_3"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i>, <a class="ref" href="jordan12.html#D2" target="_self">Def2</a>, <a class="ref" href="jordan12.html#T7" target="_self">Th7</a>, <a class="ref" href="topreal6.html#T38">TOPREAL6:38</a></i>;<br/><b>hence </b><a NAME="E6:41_3"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p</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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, , <a class="txt" href="jordan12.html#E20:41"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan12.html#E22:41"><i><font color="Green">E68</font></i></a>, , , <a class="txt" href="jordan12.html#E28:41"><i><font color="Green">E79</font></i></a>, <a class="ref" href="jordan12.html#T8" target="_self">Th8</a>, <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E163</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_4"/><i><font color="Green">E164</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><a NAME="E2:41_4"/><b>then </b>
<font color="Maroon">j1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j2</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:41_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, , <a class="txt" href="jordan12.html#E20:41"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan12.html#E22:41"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan12.html#E24:41"><i><font color="Green">E69</font></i></a>, , <a class="ref" href="jordan12.html#T8" target="_self">Th8</a>, <a class="ref" href="goboard5.html#T5">GOBOARD5:5</a></i>;<br/><a NAME="E4:41_4"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i>, <a class="ref" href="jordan12.html#D2" target="_self">Def2</a>, <a class="ref" href="jordan12.html#T7" target="_self">Th7</a>, <a class="ref" href="topreal6.html#T38">TOPREAL6:38</a></i>;<br/><b>hence </b><a NAME="E5:41_4"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p</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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, , <a class="txt" href="jordan12.html#E19:41"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan12.html#E20:41"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan12.html#E24:41"><i><font color="Green">E69</font></i></a>, <a class="ref" href="jordan12.html#T6" target="_self">Th6</a>, , , <a class="ref" href="jordan12.html#T8" target="_self">Th8</a>, <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E165</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_5"/><i><font color="Green">E166</font></i>: 
( <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><a NAME="E2:41_5"/><b>then </b>
<font color="Maroon">i2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:41_5"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j1</font></span>)</span> <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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="jordan12.html#E19:41"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan12.html#E20:41"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan12.html#E24:41"><i><font color="Green">E69</font></i></a>, <a class="ref" href="jordan12.html#T6" target="_self">Th6</a>, <a class="ref" href="goboard5.html#T4">GOBOARD5:4</a></i>;<br/><a NAME="E4:41_5"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, , , <a class="ref" href="jordan12.html#D2" target="_self">Def2</a>, <a class="ref" href="jordan12.html#T7" target="_self">Th7</a>, <a class="ref" href="jordan12.html#T9" target="_self">Th9</a>, <a class="ref" href="topreal6.html#T37">TOPREAL6:37</a></i>;<br/><b>hence </b><a NAME="E5:41_5"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p</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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, , <a class="txt" href="jordan12.html#E18:41"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan12.html#E19:41"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan12.html#E20:41"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan12.html#E22:41"><i><font color="Green">E68</font></i></a>, , , <a class="ref" href="jordan12.html#T9" target="_self">Th9</a>, <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E167</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_6"/><i><font color="Green">E168</font></i>: 
( <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><a NAME="E2:41_6"/><b>then </b>
<font color="Maroon">i1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i2</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:41_6"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, , <a class="txt" href="jordan12.html#E18:41"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan12.html#E22:41"><i><font color="Green">E68</font></i></a>, , , <a class="ref" href="jordan12.html#T10" target="_self">Th10</a>, <a class="ref" href="goboard5.html#T4">GOBOARD5:4</a></i>;<br/><a NAME="E4:41_6"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i>, <a class="ref" href="jordan12.html#D2" target="_self">Def2</a>, <a class="ref" href="jordan12.html#T7" target="_self">Th7</a>, <a class="ref" href="topreal6.html#T37">TOPREAL6:37</a></i>;<br/><b>hence </b><a NAME="E5:41_6"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p</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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, , <a class="txt" href="jordan12.html#E18:41"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan12.html#E19:41"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan12.html#E20:41"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan12.html#E22:41"><i><font color="Green">E68</font></i></a>, <a class="ref" href="jordan12.html#T6" target="_self">Th6</a>, , <a class="ref" href="jordan12.html#T10" target="_self">Th10</a>, <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E55:41"/><i><font color="Green">E169</font></i>: 
( <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 implies (  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</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">f</font></span>)</span>,<font color="Maroon">i1</font>,<font color="Maroon">j1</font></span>)</span> &amp;  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</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">f</font></span>)</span>,<font color="Maroon">i1</font>,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> ) )
 
<b>by </b><i><a class="txt" href="jordan12.html#E2"><i><font color="Green">Lemma47</font></i></a>, <a class="txt" href="jordan12.html#E3"><i><font color="Green">Lemma49</font></i></a>, <a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, , <a class="ref" href="gobrd13.html#T24">GOBRD13:24</a>, <a class="ref" href="gobrd13.html#T25">GOBRD13:25</a></i>;<br/>
<a NAME="E56:41"/><i><font color="Green">E170</font></i>: 
( <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 implies <font color="Maroon">i1</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">f</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan12.html#E18:41"><i><font color="Green">E65</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<div><i><font color="Green">E171</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_7"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_7"/>
( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j1</font> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><a NAME="E2:41_7"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">i1</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">f</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1 &amp; <font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</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">f</font></span>)</span> )
 <b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, <a class="txt" href="jordan12.html#E18:41"><i><font color="Green">E65</font></i></a>, <a class="ref" href="jordan12.html#T6" target="_self">Th6</a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a>, <a class="ref" href="jordan3.html#T12">JORDAN3:12</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E3:41_7"/>
 <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">f</font></span>)</span>,<font color="Maroon">i1</font>,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r</font>,<font color="Olive">s</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a> : ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  ) </span>}</span> 
 <b>by </b><i><a class="ref" href="goboard6.html#T29">GOBOARD6:29</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E172</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_8"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_8"/><i><font color="Green">E173</font></i>: 
( 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><a NAME="E2:41_8"/><b>then </b>
 <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">f</font></span>)</span>,<font color="Maroon">i1</font>,0</span>)</span> <a href="hidden.html#R1">=</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">f</font></span>)</span>,<font color="Maroon">i1</font>,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="nat_2.html#T10">NAT_2:10</a></i>;<br/><b>hence </b><a NAME="E3:41_8"/>
 <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">f</font></span>)</span>,<font color="Maroon">i1</font>,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r</font>,<font color="Olive">s</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a> : ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">s</font> <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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span> <a href="euclid.html#K22">`2</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jordan12.html#E22:41"><i><font color="Green">E68</font></i></a>, , , <a class="ref" href="goboard6.html#T27">GOBOARD6:27</a></i>;<br/></div><b>end;</b></div>
<a NAME="E59:41"/><i><font color="Green">E174</font></i>: 
( <font color="Maroon">j1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</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">f</font></span>)</span> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 implies  <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">f</font></span>)</span>,<font color="Maroon">i1</font>,<font color="Maroon">j1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r</font>,<font color="Olive">s</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a> : ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  ) </span>}</span>  )
 
<b>by </b><i><a class="txt" href="jordan12.html#E22:41"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan12.html#E24:41"><i><font color="Green">E69</font></i></a>, , <a class="ref" href="goboard6.html#T29">GOBOARD6:29</a></i>;<br/>
<a NAME="E60:41"/><i><font color="Green">E175</font></i>: 
( <font color="Maroon">j1</font> <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">f</font></span>)</span> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 implies  <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">f</font></span>)</span>,<font color="Maroon">i1</font>,<font color="Maroon">j1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r</font>,<font color="Olive">s</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a> : ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</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">f</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> ) </span>}</span>  )
 
<b>by </b><i><a class="txt" href="jordan12.html#E22:41"><i><font color="Green">E68</font></i></a>, , <a class="ref" href="goboard6.html#T28">GOBOARD6:28</a></i>;<br/>
<a NAME="E61:41"/><i><font color="Green">E176</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 implies <font color="Maroon">i2</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">f</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan12.html#E20:41"><i><font color="Green">E67</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E62:41"/><i><font color="Green">E177</font></i>: 
( <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 implies (  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</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">f</font></span>)</span>,<font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> &amp;  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</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">f</font></span>)</span>,<font color="Maroon">i2</font>,<font color="Maroon">j1</font></span>)</span> ) )
 
<b>by </b><i><a class="txt" href="jordan12.html#E2"><i><font color="Green">Lemma47</font></i></a>, <a class="txt" href="jordan12.html#E3"><i><font color="Green">Lemma49</font></i></a>, <a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, , <a class="ref" href="gobrd13.html#T26">GOBRD13:26</a>, <a class="ref" href="gobrd13.html#T27">GOBRD13:27</a></i>;<br/>
<div><i><font color="Green">E178</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_9"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_9"/>
( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j1</font> &amp; <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><a NAME="E2:41_9"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">i2</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">f</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1 &amp; <font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</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">f</font></span>)</span> )
 <b>by </b><i>, <a class="txt" href="jordan12.html#E20:41"><i><font color="Green">E67</font></i></a>, <a class="ref" href="jordan12.html#T6" target="_self">Th6</a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a>, <a class="ref" href="jordan3.html#T12">JORDAN3:12</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E3:41_9"/>
 <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">f</font></span>)</span>,<font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r</font>,<font color="Olive">s</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a> : ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  ) </span>}</span> 
 <b>by </b><i><a class="ref" href="goboard6.html#T29">GOBOARD6:29</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E179</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_10"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_10"/><i><font color="Green">E180</font></i>: 
( 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> &amp; <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><a NAME="E2:41_10"/><b>then </b>
 <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">f</font></span>)</span>,<font color="Maroon">i2</font>,0</span>)</span> <a href="hidden.html#R1">=</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">f</font></span>)</span>,<font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="nat_2.html#T10">NAT_2:10</a></i>;<br/><b>hence </b><a NAME="E3:41_10"/>
 <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">f</font></span>)</span>,<font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r</font>,<font color="Olive">s</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a> : ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">s</font> <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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span> <a href="euclid.html#K22">`2</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jordan12.html#E19:41"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan12.html#E30:41"><i><font color="Green">E88</font></i></a>, <a class="txt" href="jordan12.html#E36:41"><i><font color="Green">E92</font></i></a>, <a class="ref" href="goboard6.html#T27">GOBOARD6:27</a></i>;<br/></div><b>end;</b></div>
<a NAME="E65:41"/><i><font color="Green">E181</font></i>: 
( <font color="Maroon">j1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</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">f</font></span>)</span> &amp; <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 implies  <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">f</font></span>)</span>,<font color="Maroon">i2</font>,<font color="Maroon">j1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r</font>,<font color="Olive">s</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a> : ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  ) </span>}</span>  )
 
<b>by </b><i><a class="txt" href="jordan12.html#E19:41"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan12.html#E24:41"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jordan12.html#E30:41"><i><font color="Green">E88</font></i></a>, <a class="ref" href="goboard6.html#T29">GOBOARD6:29</a></i>;<br/>
<a NAME="E66:41"/><i><font color="Green">E182</font></i>: 
( <font color="Maroon">j1</font> <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">f</font></span>)</span> &amp; <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 implies  <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">f</font></span>)</span>,<font color="Maroon">i2</font>,<font color="Maroon">j1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r</font>,<font color="Olive">s</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a> : ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</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">f</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> ) </span>}</span>  )
 
<b>by </b><i><a class="txt" href="jordan12.html#E19:41"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan12.html#E30:41"><i><font color="Green">E88</font></i></a>, <a class="ref" href="goboard6.html#T28">GOBOARD6:28</a></i>;<br/>
<a NAME="E67:41"/><i><font color="Green">E183</font></i>: 
( <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 implies <font color="Maroon">j1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</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">f</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E68:41"/><i><font color="Green">E184</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 implies (  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</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">f</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j1</font></span>)</span> &amp;  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</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">f</font></span>)</span>,<font color="Maroon">i1</font>,<font color="Maroon">j1</font></span>)</span> ) )
 
<b>by </b><i><a class="txt" href="jordan12.html#E2"><i><font color="Green">Lemma47</font></i></a>, <a class="txt" href="jordan12.html#E3"><i><font color="Green">Lemma49</font></i></a>, <a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, , <a class="ref" href="gobrd13.html#T22">GOBRD13:22</a>, <a class="ref" href="gobrd13.html#T23">GOBRD13:23</a></i>;<br/>
<div><i><font color="Green">E185</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_11"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_11"/>
( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><a NAME="E2:41_11"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1 &amp; <font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1 <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">f</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j1</font> &amp; <font color="Maroon">j1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</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">f</font></span>)</span> )
 <b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, <a class="txt" href="jordan12.html#E20:41"><i><font color="Green">E67</font></i></a>, , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a>, <a class="ref" href="jordan3.html#T12">JORDAN3:12</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E3:41_11"/>
 <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">f</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r</font>,<font color="Olive">s</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a> : ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  ) </span>}</span> 
 <b>by </b><i><a class="ref" href="goboard6.html#T29">GOBOARD6:29</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E186</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_12"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_12"/><i><font color="Green">E187</font></i>: 
( 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><a NAME="E2:41_12"/><b>then </b>
 <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">f</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j1</font></span>)</span> <a href="hidden.html#R1">=</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">f</font></span>)</span>,0,<font color="Maroon">j1</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_2.html#T10">NAT_2:10</a></i>;<br/><b>hence </b><a NAME="E3:41_12"/>
 <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">f</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r</font>,<font color="Olive">s</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Olive">r</font> <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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jordan12.html#E24:41"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jordan12.html#E38:41"><i><font color="Green">E94</font></i></a>, , <a class="ref" href="goboard6.html#T23">GOBOARD6:23</a></i>;<br/></div><b>end;</b></div>
<a NAME="E71:41"/><i><font color="Green">E188</font></i>: 
( <font color="Maroon">i1</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">f</font></span>)</span> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 implies  <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">f</font></span>)</span>,<font color="Maroon">i1</font>,<font color="Maroon">j1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r</font>,<font color="Olive">s</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a> : ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  ) </span>}</span>  )
 
<b>by </b><i><a class="txt" href="jordan12.html#E22:41"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan12.html#E24:41"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jordan12.html#E38:41"><i><font color="Green">E94</font></i></a>, <a class="ref" href="goboard6.html#T29">GOBOARD6:29</a></i>;<br/>
<a NAME="E72:41"/><i><font color="Green">E189</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</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">f</font></span>)</span> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 implies  <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">f</font></span>)</span>,<font color="Maroon">i1</font>,<font color="Maroon">j1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r</font>,<font color="Olive">s</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a> : ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</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">f</font></span>)</span></span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  ) </span>}</span>  )
 
<b>by </b><i><a class="txt" href="jordan12.html#E24:41"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jordan12.html#E38:41"><i><font color="Green">E94</font></i></a>, <a class="ref" href="goboard6.html#T26">GOBOARD6:26</a></i>;<br/>
<a NAME="E73:41"/><i><font color="Green">E190</font></i>: 
( <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 implies <font color="Maroon">j2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</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">f</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="jordan12.html#T6" target="_self">Th6</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E74:41"/><i><font color="Green">E191</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 implies (  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</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">f</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span> &amp;  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</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">f</font></span>)</span>,<font color="Maroon">i2</font>,<font color="Maroon">j2</font></span>)</span> ) )
 
<b>by </b><i><a class="txt" href="jordan12.html#E2"><i><font color="Green">Lemma47</font></i></a>, <a class="txt" href="jordan12.html#E3"><i><font color="Green">Lemma49</font></i></a>, <a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, , <a class="ref" href="gobrd13.html#T28">GOBRD13:28</a>, <a class="ref" href="gobrd13.html#T29">GOBRD13:29</a></i>;<br/>
<div><i><font color="Green">E192</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_13"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_13"/>
( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><a NAME="E2:41_13"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1 &amp; <font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1 <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">f</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">j2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</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">f</font></span>)</span> )
 <b>by </b><i>, <a class="txt" href="jordan12.html#E18:41"><i><font color="Green">E65</font></i></a>, <a class="ref" href="jordan12.html#T6" target="_self">Th6</a>, <a class="ref" href="jordan12.html#D1" target="_self">Def1</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a>, <a class="ref" href="jordan3.html#T12">JORDAN3:12</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E3:41_13"/>
 <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">f</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r</font>,<font color="Olive">s</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a> : ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  ) </span>}</span> 
 <b>by </b><i><a class="ref" href="goboard6.html#T29">GOBOARD6:29</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E193</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_14"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_14"/><i><font color="Green">E194</font></i>: 
( 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><a NAME="E2:41_14"/><b>then </b>
 <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">f</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span> <a href="hidden.html#R1">=</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">f</font></span>)</span>,0,<font color="Maroon">j2</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_2.html#T10">NAT_2:10</a></i>;<br/><b>hence </b><a NAME="E3:41_14"/>
 <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">f</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r</font>,<font color="Olive">s</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Olive">r</font> <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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  ) </span>}</span> 
 <b>by </b><i>, , , <a class="ref" href="goboard6.html#T23">GOBOARD6:23</a></i>;<br/></div><b>end;</b></div>
<a NAME="E77:41"/><i><font color="Green">E195</font></i>: 
( <font color="Maroon">i2</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">f</font></span>)</span> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 implies  <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">f</font></span>)</span>,<font color="Maroon">i2</font>,<font color="Maroon">j2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r</font>,<font color="Olive">s</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a> : ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  ) </span>}</span>  )
 
<b>by </b><i><a class="txt" href="jordan12.html#E19:41"><i><font color="Green">E66</font></i></a>, , , <a class="ref" href="goboard6.html#T29">GOBOARD6:29</a></i>;<br/>
<a NAME="E78:41"/><i><font color="Green">E196</font></i>: 
( <font color="Maroon">i2</font> <a href="hidden.html#R1">=</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">f</font></span>)</span> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 implies  <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">f</font></span>)</span>,<font color="Maroon">i2</font>,<font color="Maroon">j2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r</font>,<font color="Olive">s</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a> : ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</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">f</font></span>)</span></span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  ) </span>}</span>  )
 
<b>by </b><i>, , <a class="ref" href="goboard6.html#T26">GOBOARD6:26</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_15"/>
( <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  or <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_1"><b>suppose </b></a><a NAME="E1:41_15_1"/><i><font color="Green">E197</font></i>: 
<font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> 
 ;<br/><div class="add"><a NAME="E2:41_15_1"/><b>then </b><i><font color="Green">E198</font></i>: 
<font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, <a class="ref" href="goboard7.html#T6">GOBOARD7:6</a></i>;<br/><a NAME="E3:41_15_1"/>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="jordan12.html#E2"><i><font color="Green">Lemma47</font></i></a>, <a class="txt" href="jordan12.html#E3"><i><font color="Green">Lemma49</font></i></a>, <a class="ref" href="jordan9.html#T11">JORDAN9:11</a></i>;<br/><b>then consider </b><font color="Maroon">rl'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:41_15_1"/><i><font color="Green">E200</font></i>: 
<font color="Maroon">rl'</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">rl'</font> = <font color="Maroon">rl'</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="jordan12.html#T14" target="_self">Th14</a></i>;<br/><b>reconsider </b><font color="Maroon">rl</font> = <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">rl'</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E8:41_15_1"/><i><font color="Green">E201</font></i>: 
( <font color="Maroon">rl</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">rl</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">rl'</font> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E9:41_15_1"/>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="jordan12.html#E2"><i><font color="Green">Lemma47</font></i></a>, <a class="txt" href="jordan12.html#E3"><i><font color="Green">Lemma49</font></i></a>, <a class="ref" href="jordan9.html#T11">JORDAN9:11</a></i>;<br/><b>then consider </b><font color="Maroon">rp'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:41_15_1"/><i><font color="Green">E203</font></i>: 
<font color="Maroon">rp'</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">rp'</font> = <font color="Maroon">rp'</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="jordan12.html#T16" target="_self">Th16</a></i>;<br/><b>reconsider </b><font color="Maroon">rp</font> = <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">rp'</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E14:41_15_1"/><i><font color="Green">E204</font></i>: 
( <font color="Maroon">rp</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">rp</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">rp'</font> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><div><i><font color="Green">E205</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_15_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_15_1_1"/><i><font color="Green">E206</font></i>: 
( <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">i2</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">f</font></span>)</span> )
 ;<br/><a NAME="E2:41_15_1_1"/><b>then </b>
 ex <font color="Olive">r</font>, <font color="Olive">s</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">rl'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r</font>,<font color="Olive">s</font></span><a href="euclid.html#K23">]|</a></span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, , <a class="txt" href="jordan12.html#E14:41"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan12.html#E15:41"><i><font color="Green">E61</font></i></a>, , <a class="ref" href="jordan12.html#T12" target="_self">Th12</a>, , , <a class="ref" href="jordan12.html#T14" target="_self">Th14</a>, , <a class="ref" href="jordan1g.html#T7">JORDAN1G:7</a></i>;<br/><a NAME="E3:41_15_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">rl</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">rl</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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p</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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, , <a class="txt" href="jordan12.html#E14:41"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan12.html#E15:41"><i><font color="Green">E61</font></i></a>, , <a class="txt" href="jordan12.html#E26:41"><i><font color="Green">E78</font></i></a>, , <a class="ref" href="jordan12.html#T15" target="_self">Th15</a>, , , <a class="ref" href="euclid.html#T56">EUCLID:56</a>, <a class="ref" href="jordan1g.html#T7">JORDAN1G:7</a></i>;<br/><b>hence </b><a NAME="E4:41_15_1_1"/>
<font color="Maroon">rl</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, , <a class="txt" href="jordan12.html#E14:41"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan12.html#E15:41"><i><font color="Green">E61</font></i></a>, , <a class="ref" href="jordan12.html#T12" target="_self">Th12</a>, , , <a class="ref" href="jordan12.html#T15" target="_self">Th15</a>, , , <a class="ref" href="jordan1g.html#T7">JORDAN1G:7</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E207</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_15_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_15_1_2"/><i><font color="Green">E208</font></i>: 
( <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">i2</font> <a href="hidden.html#R1">=</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">f</font></span>)</span> )
 ;<br/><a NAME="E2:41_15_1_2"/><b>then </b>
 ex <font color="Olive">r</font>, <font color="Olive">s</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">rl'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r</font>,<font color="Olive">s</font></span><a href="euclid.html#K23">]|</a></span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</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">f</font></span>)</span></span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, , <a class="txt" href="jordan12.html#E14:41"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan12.html#E15:41"><i><font color="Green">E61</font></i></a>, , <a class="ref" href="jordan12.html#T12" target="_self">Th12</a>, , , <a class="ref" href="jordan12.html#T14" target="_self">Th14</a>, , <a class="ref" href="jordan1g.html#T7">JORDAN1G:7</a></i>;<br/><a NAME="E3:41_15_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</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">f</font></span>)</span></span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">rl</font> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p</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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, , <a class="txt" href="jordan12.html#E14:41"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan12.html#E15:41"><i><font color="Green">E61</font></i></a>, , <a class="txt" href="jordan12.html#E26:41"><i><font color="Green">E78</font></i></a>, , <a class="ref" href="jordan12.html#T15" target="_self">Th15</a>, <a class="txt" href="jordan12.html#E24"><i><font color="Green">Lemma114</font></i></a>, , <a class="ref" href="euclid.html#T56">EUCLID:56</a>, <a class="ref" href="jordan1g.html#T7">JORDAN1G:7</a></i>;<br/><b>hence </b><a NAME="E4:41_15_1_2"/>
<font color="Maroon">rl</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, , <a class="txt" href="jordan12.html#E14:41"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan12.html#E15:41"><i><font color="Green">E61</font></i></a>, , <a class="ref" href="jordan12.html#T12" target="_self">Th12</a>, , , <a class="ref" href="jordan12.html#T15" target="_self">Th15</a>, <a class="txt" href="jordan12.html#E24"><i><font color="Green">Lemma114</font></i></a>, , <a class="ref" href="jordan1g.html#T7">JORDAN1G:7</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E209</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_15_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_15_1_3"/><i><font color="Green">E210</font></i>: 
( <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 &amp; 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> )
 ;<br/><a NAME="E2:41_15_1_3"/><b>then </b>
 ex <font color="Olive">r</font>, <font color="Olive">s</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">rp'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r</font>,<font color="Olive">s</font></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Olive">r</font> <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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, , <a class="txt" href="jordan12.html#E14:41"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan12.html#E15:41"><i><font color="Green">E61</font></i></a>, , <a class="ref" href="jordan12.html#T12" target="_self">Th12</a>, <a class="ref" href="jordan12.html#T13" target="_self">Th13</a>, , <a class="ref" href="jordan12.html#T16" target="_self">Th16</a>, , <a class="ref" href="jordan1g.html#T7">JORDAN1G:7</a></i>;<br/><a NAME="E3:41_15_1_3"/><b>then </b>
<font color="Maroon">rp'</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">f</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="euclid.html#T56">EUCLID:56</a></i>;<br/><b>hence </b><a NAME="E4:41_15_1_3"/>
<font color="Maroon">rp</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, , <a class="txt" href="jordan12.html#E14:41"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan12.html#E15:41"><i><font color="Green">E61</font></i></a>, , <a class="txt" href="jordan12.html#E26:41"><i><font color="Green">E78</font></i></a>, <a class="ref" href="jordan12.html#T12" target="_self">Th12</a>, <a class="ref" href="jordan12.html#T13" target="_self">Th13</a>, , , , <a class="ref" href="jordan1g.html#T7">JORDAN1G:7</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E211</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_15_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_15_1_4"/><i><font color="Green">E212</font></i>: 
( <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 &amp; 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i2</font> )
 ;<br/><a NAME="E2:41_15_1_4"/><b>then </b>
 ex <font color="Olive">r</font>, <font color="Olive">s</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">rp'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r</font>,<font color="Olive">s</font></span><a href="euclid.html#K23">]|</a></span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, , <a class="txt" href="jordan12.html#E14:41"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan12.html#E15:41"><i><font color="Green">E61</font></i></a>, , <a class="ref" href="jordan12.html#T12" target="_self">Th12</a>, , , <a class="ref" href="jordan12.html#T16" target="_self">Th16</a>, , <a class="ref" href="jordan1g.html#T7">JORDAN1G:7</a></i>;<br/><a NAME="E3:41_15_1_4"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">rp'</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">rp'</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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><b>hence </b><a NAME="E4:41_15_1_4"/>
<font color="Maroon">rp</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, , <a class="txt" href="jordan12.html#E14:41"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan12.html#E15:41"><i><font color="Green">E61</font></i></a>, , <a class="txt" href="jordan12.html#E26:41"><i><font color="Green">E78</font></i></a>, <a class="ref" href="jordan12.html#T12" target="_self">Th12</a>, , , , , <a class="ref" href="jordan1g.html#T7">JORDAN1G:7</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E213</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_15_1_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_15_1_5"/><i><font color="Green">E214</font></i>: 
( <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">i1</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">f</font></span>)</span> )
 ;<br/><a NAME="E2:41_15_1_5"/><b>then </b>
 ex <font color="Olive">r</font>, <font color="Olive">s</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">rp'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r</font>,<font color="Olive">s</font></span><a href="euclid.html#K23">]|</a></span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, , <a class="txt" href="jordan12.html#E14:41"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan12.html#E15:41"><i><font color="Green">E61</font></i></a>, , <a class="ref" href="jordan12.html#T11" target="_self">Th11</a>, , , <a class="ref" href="jordan12.html#T16" target="_self">Th16</a>, , <a class="ref" href="jordan1g.html#T7">JORDAN1G:7</a></i>;<br/><a NAME="E3:41_15_1_5"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">rp'</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">rp'</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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><b>hence </b><a NAME="E4:41_15_1_5"/>
<font color="Maroon">rp</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, , <a class="txt" href="jordan12.html#E14:41"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan12.html#E15:41"><i><font color="Green">E61</font></i></a>, , <a class="txt" href="jordan12.html#E28:41"><i><font color="Green">E79</font></i></a>, <a class="ref" href="jordan12.html#T11" target="_self">Th11</a>, , , , , <a class="ref" href="jordan1g.html#T7">JORDAN1G:7</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E215</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_15_1_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_15_1_6"/><i><font color="Green">E216</font></i>: 
( <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">i1</font> <a href="hidden.html#R1">=</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">f</font></span>)</span> )
 ;<br/><a NAME="E2:41_15_1_6"/><b>then </b>
 ex <font color="Olive">r</font>, <font color="Olive">s</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">rp'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r</font>,<font color="Olive">s</font></span><a href="euclid.html#K23">]|</a></span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</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">f</font></span>)</span></span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, , <a class="txt" href="jordan12.html#E14:41"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan12.html#E15:41"><i><font color="Green">E61</font></i></a>, , <a class="ref" href="jordan12.html#T11" target="_self">Th11</a>, , , <a class="ref" href="jordan12.html#T16" target="_self">Th16</a>, , <a class="ref" href="jordan1g.html#T7">JORDAN1G:7</a></i>;<br/><a NAME="E3:41_15_1_6"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</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">f</font></span>)</span></span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">rp'</font> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><b>hence </b><a NAME="E4:41_15_1_6"/>
<font color="Maroon">rp</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, , <a class="txt" href="jordan12.html#E14:41"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan12.html#E15:41"><i><font color="Green">E61</font></i></a>, , <a class="txt" href="jordan12.html#E28:41"><i><font color="Green">E79</font></i></a>, <a class="ref" href="jordan12.html#T11" target="_self">Th11</a>, , , , , <a class="ref" href="jordan1g.html#T7">JORDAN1G:7</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E217</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_15_1_7"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_15_1_7"/><i><font color="Green">E218</font></i>: 
( <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 &amp; 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> )
 ;<br/><a NAME="E2:41_15_1_7"/><b>then </b>
 ex <font color="Olive">r</font>, <font color="Olive">s</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">rl'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r</font>,<font color="Olive">s</font></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Olive">r</font> <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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, , <a class="txt" href="jordan12.html#E14:41"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan12.html#E15:41"><i><font color="Green">E61</font></i></a>, , <a class="ref" href="jordan12.html#T11" target="_self">Th11</a>, , , <a class="ref" href="jordan12.html#T14" target="_self">Th14</a>, , <a class="ref" href="jordan1g.html#T7">JORDAN1G:7</a></i>;<br/><a NAME="E3:41_15_1_7"/><b>then </b>
<font color="Maroon">rl'</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">f</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="euclid.html#T56">EUCLID:56</a></i>;<br/><b>hence </b><a NAME="E4:41_15_1_7"/>
<font color="Maroon">rl</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, , <a class="txt" href="jordan12.html#E14:41"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan12.html#E15:41"><i><font color="Green">E61</font></i></a>, , <a class="txt" href="jordan12.html#E28:41"><i><font color="Green">E79</font></i></a>, <a class="ref" href="jordan12.html#T11" target="_self">Th11</a>, , , , , <a class="ref" href="jordan1g.html#T7">JORDAN1G:7</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E219</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_15_1_8"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_15_1_8"/><i><font color="Green">E220</font></i>: 
( <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 &amp; 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i1</font> )
 ;<br/><a NAME="E2:41_15_1_8"/><b>then </b>
 ex <font color="Olive">r</font>, <font color="Olive">s</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">rl'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r</font>,<font color="Olive">s</font></span><a href="euclid.html#K23">]|</a></span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, , <a class="txt" href="jordan12.html#E14:41"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan12.html#E15:41"><i><font color="Green">E61</font></i></a>, , <a class="ref" href="jordan12.html#T11" target="_self">Th11</a>, , , <a class="ref" href="jordan12.html#T14" target="_self">Th14</a>, , <a class="ref" href="jordan1g.html#T7">JORDAN1G:7</a></i>;<br/><a NAME="E3:41_15_1_8"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">rl'</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">rl'</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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><b>hence </b><a NAME="E4:41_15_1_8"/>
<font color="Maroon">rl</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, , <a class="txt" href="jordan12.html#E14:41"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan12.html#E15:41"><i><font color="Green">E61</font></i></a>, , <a class="txt" href="jordan12.html#E28:41"><i><font color="Green">E79</font></i></a>, <a class="ref" href="jordan12.html#T11" target="_self">Th11</a>, , , , , <a class="ref" href="jordan1g.html#T7">JORDAN1G:7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_15_1_9"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_1_9_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_15_1_9_1"/>
( <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 )
 </span><b>by </b><i><a class="txt" href="jordan12.html#E4:41"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan12.html#E16:41"><i><font color="Green">E63</font></i></a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_1_9_1_1"><b>suppose </b></a><a NAME="E1:41_15_1_9_1_1"/><i><font color="Green">E221</font></i>: 
<font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:41_15_1_9_1_1"/><i><font color="Green">E222</font></i>: 
<font color="Maroon">rl</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_15_1_9_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_1_9_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_15_1_9_1_1_1_1"/>
( <font color="Maroon">i2</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">f</font></span>)</span> or <font color="Maroon">i2</font> <a href="hidden.html#R1">=</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">f</font></span>)</span> )
 </span><b>by </b><i><a class="txt" href="jordan12.html#E18:41"><i><font color="Green">E65</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_1_9_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:41_15_1_9_1_1_1_1_1"/>
<font color="Maroon">i2</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">f</font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_15_1_9_1_1_1_1_1"/>
<font color="Maroon">rl</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan12.html#T18" target="_self">Th18</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_1_9_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:41_15_1_9_1_1_1_1_2"/>
<font color="Maroon">i2</font> <a href="hidden.html#R1">=</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">f</font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_15_1_9_1_1_1_1_2"/>
<font color="Maroon">rl</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:41_15_1_9_1_1"/><i><font color="Green">E223</font></i>: 
<font color="Maroon">rp</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_15_1_9_1_1_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_1_9_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_15_1_9_1_1_2_1"/>
( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i2</font> or 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> )
 </span><b>by </b><i><a class="txt" href="jordan12.html#E19:41"><i><font color="Green">E66</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_1_9_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:41_15_1_9_1_1_2_1_1"/>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_15_1_9_1_1_2_1_1"/>
<font color="Maroon">rp</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_1_9_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:41_15_1_9_1_1_2_1_2"/>
1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_15_1_9_1_1_2_1_2"/>
<font color="Maroon">rp</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E24"><i><font color="Green">Lemma114</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E4:41_15_1_9_1_1"/><i><font color="Green">E224</font></i>: 
<font color="Maroon">rl</font> <a href="hidden.html#R2">in</a>  <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan12.html#T4" target="_self">Th4</a>, <a class="ref" href="jordan12.html#T19" target="_self">Th19</a></i>;<br/><a NAME="E5:41_15_1_9_1_1"/><i><font color="Green">E225</font></i>: 
<font color="Maroon">rp</font> <a href="hidden.html#R2">in</a>  <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan12.html#T3" target="_self">Th3</a>, </i>;<br/><a NAME="E6:41_15_1_9_1_1"/><i><font color="Green">E226</font></i>: 
not <font color="Maroon">rl</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="jordan12.html#T4" target="_self">Th4</a>, <a class="ref" href="jordan12.html#T19" target="_self">Th19</a></i>;<br/><a NAME="E7:41_15_1_9_1_1"/>
not <font color="Maroon">rp</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="jordan12.html#T3" target="_self">Th3</a>, </i>;<br/><b>hence </b><a NAME="E8:41_15_1_9_1_1"/>
for <font color="Olive">C</font> being  <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>  holds <br/>(  not <font color="Olive">C</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">f</font></span>)</span> <a href="subset_1.html#K3">`</a>  or  not <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> or  not <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 <b>by </b><i><a class="txt" href="jordan12.html#E1"><i><font color="Green">Lemma40</font></i></a>, <a class="txt" href="jordan12.html#E1:41"><i><font color="Green">E42</font></i></a>, , <a class="txt" href="jordan12.html#E3"><i><font color="Green">Lemma49</font></i></a>, , , <a class="ref" href="jordan12.html#T15" target="_self">Th15</a>, <a class="ref" href="jordan12.html#T17" target="_self">Th17</a>, <a class="txt" href="jordan12.html#E26"><i><font color="Green">Lemma123</font></i></a>, <a class="ref" href="jordan12.html#T20" target="_self">Th20</a>, <a class="ref" href="jordan12.html#T21" target="_self">Th21</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_1_9_1_2"><b>suppose </b></a><a NAME="E1:41_15_1_9_1_2"/><i><font color="Green">E227</font></i>: 
<font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:41_15_1_9_1_2"/><i><font color="Green">E228</font></i>: 
<font color="Maroon">rl</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_15_1_9_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_1_9_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_15_1_9_1_2_1_1"/>
( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i1</font> or 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> )
 </span><b>by </b><i><a class="txt" href="jordan12.html#E22:41"><i><font color="Green">E68</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_1_9_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:41_15_1_9_1_2_1_1_1"/>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_15_1_9_1_2_1_1_1"/>
<font color="Maroon">rl</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_1_9_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:41_15_1_9_1_2_1_1_2"/>
1 <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_15_1_9_1_2_1_1_2"/>
<font color="Maroon">rl</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:41_15_1_9_1_2"/><i><font color="Green">E229</font></i>: 
<font color="Maroon">rp</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_15_1_9_1_2_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_1_9_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_15_1_9_1_2_2_1"/>
( <font color="Maroon">i1</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">f</font></span>)</span> or <font color="Maroon">i1</font> <a href="hidden.html#R1">=</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">f</font></span>)</span> )
 </span><b>by </b><i><a class="txt" href="jordan12.html#E20:41"><i><font color="Green">E67</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_1_9_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:41_15_1_9_1_2_2_1_1"/>
<font color="Maroon">i1</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">f</font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_15_1_9_1_2_2_1_1"/>
<font color="Maroon">rp</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_1_9_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:41_15_1_9_1_2_2_1_2"/>
<font color="Maroon">i1</font> <a href="hidden.html#R1">=</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">f</font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_15_1_9_1_2_2_1_2"/>
<font color="Maroon">rp</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E4:41_15_1_9_1_2"/><i><font color="Green">E230</font></i>: 
<font color="Maroon">rl</font> <a href="hidden.html#R2">in</a>  <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan12.html#T4" target="_self">Th4</a>, <a class="ref" href="jordan12.html#T19" target="_self">Th19</a></i>;<br/><a NAME="E5:41_15_1_9_1_2"/><i><font color="Green">E231</font></i>: 
<font color="Maroon">rp</font> <a href="hidden.html#R2">in</a>  <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan12.html#T3" target="_self">Th3</a>, </i>;<br/><a NAME="E6:41_15_1_9_1_2"/><i><font color="Green">E232</font></i>: 
not <font color="Maroon">rl</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="jordan12.html#T4" target="_self">Th4</a>, <a class="ref" href="jordan12.html#T19" target="_self">Th19</a></i>;<br/><a NAME="E7:41_15_1_9_1_2"/>
not <font color="Maroon">rp</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="jordan12.html#T3" target="_self">Th3</a>, </i>;<br/><b>hence </b><a NAME="E8:41_15_1_9_1_2"/>
for <font color="Olive">C</font> being  <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>  holds <br/>(  not <font color="Olive">C</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">f</font></span>)</span> <a href="subset_1.html#K3">`</a>  or  not <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> or  not <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 <b>by </b><i><a class="txt" href="jordan12.html#E1"><i><font color="Green">Lemma40</font></i></a>, <a class="txt" href="jordan12.html#E1:41"><i><font color="Green">E42</font></i></a>, , <a class="txt" href="jordan12.html#E3"><i><font color="Green">Lemma49</font></i></a>, , , <a class="ref" href="jordan12.html#T15" target="_self">Th15</a>, <a class="ref" href="jordan12.html#T17" target="_self">Th17</a>, <a class="txt" href="jordan12.html#E26"><i><font color="Green">Lemma123</font></i></a>, <a class="ref" href="jordan12.html#T20" target="_self">Th20</a>, <a class="ref" href="jordan12.html#T21" target="_self">Th21</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E24:41_15_1"/>
for <font color="Olive">C</font> being  <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>  holds <br/>(  not <font color="Olive">C</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">f</font></span>)</span> <a href="subset_1.html#K3">`</a>  or  not <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> or  not <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_2"><b>suppose </b></a><a NAME="E1:41_15_2"/><i><font color="Green">E233</font></i>: 
<font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> 
 ;<br/><div class="add"><a NAME="E2:41_15_2"/><b>then </b><i><font color="Green">E234</font></i>: 
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, <a class="ref" href="goboard7.html#T5">GOBOARD7:5</a></i>;<br/><a NAME="E3:41_15_2"/>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="jordan12.html#E2"><i><font color="Green">Lemma47</font></i></a>, <a class="txt" href="jordan12.html#E3"><i><font color="Green">Lemma49</font></i></a>, <a class="ref" href="jordan9.html#T11">JORDAN9:11</a></i>;<br/><b>then consider </b><font color="Maroon">rl'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:41_15_2"/><i><font color="Green">E235</font></i>: 
<font color="Maroon">rl'</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">rl'</font> = <font color="Maroon">rl'</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="jordan12.html#T14" target="_self">Th14</a></i>;<br/><b>reconsider </b><font color="Maroon">rl</font> = <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">rl'</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E8:41_15_2"/><i><font color="Green">E236</font></i>: 
( <font color="Maroon">rl</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">rl</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">rl'</font> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E9:41_15_2"/>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="jordan12.html#E2"><i><font color="Green">Lemma47</font></i></a>, <a class="txt" href="jordan12.html#E3"><i><font color="Green">Lemma49</font></i></a>, <a class="ref" href="jordan9.html#T11">JORDAN9:11</a></i>;<br/><b>then consider </b><font color="Maroon">rp'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:41_15_2"/><i><font color="Green">E237</font></i>: 
<font color="Maroon">rp'</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">rp'</font> = <font color="Maroon">rp'</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="jordan12.html#T16" target="_self">Th16</a></i>;<br/><b>reconsider </b><font color="Maroon">rp</font> = <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">rp'</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E14:41_15_2"/><i><font color="Green">E238</font></i>: 
( <font color="Maroon">rp</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">rp</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">rp'</font> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><div><i><font color="Green">E239</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_15_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_15_2_1"/><i><font color="Green">E240</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</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">f</font></span>)</span> )
 ;<br/><a NAME="E2:41_15_2_1"/><b>then </b>
 ex <font color="Olive">r</font>, <font color="Olive">s</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">rp'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r</font>,<font color="Olive">s</font></span><a href="euclid.html#K23">]|</a></span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, , <a class="txt" href="jordan12.html#E14:41"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan12.html#E15:41"><i><font color="Green">E61</font></i></a>, , <a class="txt" href="jordan12.html#E33:41"><i><font color="Green">E89</font></i></a>, <a class="txt" href="jordan12.html#E36:41"><i><font color="Green">E92</font></i></a>, , <a class="ref" href="jordan12.html#T16" target="_self">Th16</a>, , <a class="ref" href="jordan1g.html#T6">JORDAN1G:6</a></i>;<br/><a NAME="E3:41_15_2_1"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">rp</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">rp</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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="ref" href="jordan12.html#T17" target="_self">Th17</a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><b>hence </b><a NAME="E4:41_15_2_1"/>
<font color="Maroon">rp</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, , <a class="txt" href="jordan12.html#E14:41"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan12.html#E15:41"><i><font color="Green">E61</font></i></a>, , <a class="ref" href="jordan12.html#T8" target="_self">Th8</a>, <a class="txt" href="jordan12.html#E33:41"><i><font color="Green">E89</font></i></a>, <a class="txt" href="jordan12.html#E36:41"><i><font color="Green">E92</font></i></a>, , <a class="ref" href="jordan12.html#T17" target="_self">Th17</a>, , , <a class="ref" href="jordan1g.html#T6">JORDAN1G:6</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E241</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_15_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_15_2_2"/><i><font color="Green">E242</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <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">f</font></span>)</span> )
 ;<br/><a NAME="E2:41_15_2_2"/><b>then </b>
 ex <font color="Olive">r</font>, <font color="Olive">s</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">rp'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r</font>,<font color="Olive">s</font></span><a href="euclid.html#K23">]|</a></span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</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">f</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> )
 <b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, , <a class="txt" href="jordan12.html#E14:41"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan12.html#E15:41"><i><font color="Green">E61</font></i></a>, , <a class="txt" href="jordan12.html#E33:41"><i><font color="Green">E89</font></i></a>, <a class="txt" href="jordan12.html#E37:41"><i><font color="Green">E93</font></i></a>, , <a class="ref" href="jordan12.html#T16" target="_self">Th16</a>, , <a class="ref" href="jordan1g.html#T6">JORDAN1G:6</a></i>;<br/><a NAME="E3:41_15_2_2"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p</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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</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">f</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">rp</font> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, , <a class="txt" href="jordan12.html#E14:41"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan12.html#E15:41"><i><font color="Green">E61</font></i></a>, , <a class="ref" href="jordan12.html#T8" target="_self">Th8</a>, , <a class="ref" href="jordan12.html#T17" target="_self">Th17</a>, <a class="txt" href="jordan12.html#E24"><i><font color="Green">Lemma114</font></i></a>, , <a class="ref" href="euclid.html#T56">EUCLID:56</a>, <a class="ref" href="jordan1g.html#T6">JORDAN1G:6</a></i>;<br/><b>hence </b><a NAME="E4:41_15_2_2"/>
<font color="Maroon">rp</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, , <a class="txt" href="jordan12.html#E14:41"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan12.html#E15:41"><i><font color="Green">E61</font></i></a>, , <a class="txt" href="jordan12.html#E33:41"><i><font color="Green">E89</font></i></a>, <a class="txt" href="jordan12.html#E37:41"><i><font color="Green">E93</font></i></a>, , <a class="ref" href="jordan12.html#T17" target="_self">Th17</a>, <a class="txt" href="jordan12.html#E24"><i><font color="Green">Lemma114</font></i></a>, , <a class="ref" href="jordan1g.html#T6">JORDAN1G:6</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E243</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_15_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_15_2_3"/><i><font color="Green">E244</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> )
 ;<br/><a NAME="E2:41_15_2_3"/><b>then </b>
 ex <font color="Olive">r</font>, <font color="Olive">s</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">rl'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r</font>,<font color="Olive">s</font></span><a href="euclid.html#K23">]|</a></span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">s</font> <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">f</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="jordan12.html#T5" target="_self">Th5</a>, , <a class="txt" href="jordan12.html#E14:41"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan12.html#E15:41"><i><font color="Green">E61</font></i></a>, , <a class="txt" href="jordan12.html#E33:41"><i><font color="Green">E89</font></i></a>, <a class="txt" href="jordan12.html#E35:41"><i><font color="Green">E91</font></i></a>, , <a class="ref" href="jordan12.html#T14" target="_self">Th14</a>, , <a class="ref" href="jordan1g.html#T6">JORDAN1G:6</a></i>;<br/><a NAME="E3:41_15_2_3"/><b>then </b>
<font color="Maroon">rl</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">f</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="jordan12.html#T15" target="_self">Th15</a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><b>hence </b><a NAME="E4:41_15_2_3"/>
<font color="Maroon">rl</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, , <a class="txt" href="jordan12.html#E14:41"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan12.html#E15:41"><i><font color="Green">E61</font></i></a>, , <a class="ref" href="jordan12.html#T8" target="_self">Th8</a>, <a class="txt" href="jordan12.html#E33:41"><i><font color="Green">E89</font></i></a>, <a class="txt" href="jordan12.html#E35:41"><i><font color="Green">E91</font></i></a>, , <a class="ref" href="jordan12.html#T15" target="_self">Th15</a>, , , <a class="ref" href="jordan1g.html#T6">JORDAN1G:6</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E245</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_15_2_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_15_2_4"/><i><font color="Green">E246</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j1</font> )
 ;<br/><a NAME="E2:41_15_2_4"/><b>then </b>
 ex <font color="Olive">r</font>, <font color="Olive">s</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">rl'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r</font>,<font color="Olive">s</font></span><a href="euclid.html#K23">]|</a></span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, , <a class="txt" href="jordan12.html#E14:41"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan12.html#E15:41"><i><font color="Green">E61</font></i></a>, , <a class="txt" href="jordan12.html#E33:41"><i><font color="Green">E89</font></i></a>, <a class="txt" href="jordan12.html#E34:41"><i><font color="Green">E90</font></i></a>, , <a class="ref" href="jordan12.html#T14" target="_self">Th14</a>, , <a class="ref" href="jordan1g.html#T6">JORDAN1G:6</a></i>;<br/><a NAME="E3:41_15_2_4"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">rl'</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">rl'</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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><b>hence </b><a NAME="E4:41_15_2_4"/>
<font color="Maroon">rl</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, , <a class="txt" href="jordan12.html#E14:41"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan12.html#E15:41"><i><font color="Green">E61</font></i></a>, , <a class="ref" href="jordan12.html#T8" target="_self">Th8</a>, <a class="txt" href="jordan12.html#E33:41"><i><font color="Green">E89</font></i></a>, <a class="txt" href="jordan12.html#E34:41"><i><font color="Green">E90</font></i></a>, , , , <a class="ref" href="jordan1g.html#T6">JORDAN1G:6</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E247</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_15_2_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_15_2_5"/><i><font color="Green">E248</font></i>: 
( <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</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">f</font></span>)</span> )
 ;<br/><a NAME="E2:41_15_2_5"/><b>then </b>
 ex <font color="Olive">r</font>, <font color="Olive">s</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">rl'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r</font>,<font color="Olive">s</font></span><a href="euclid.html#K23">]|</a></span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, , <a class="txt" href="jordan12.html#E14:41"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan12.html#E15:41"><i><font color="Green">E61</font></i></a>, , <a class="ref" href="jordan12.html#T10" target="_self">Th10</a>, , , <a class="ref" href="jordan12.html#T14" target="_self">Th14</a>, , <a class="ref" href="jordan1g.html#T6">JORDAN1G:6</a></i>;<br/><a NAME="E3:41_15_2_5"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">rl</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">rl</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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="ref" href="jordan12.html#T15" target="_self">Th15</a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><b>hence </b><a NAME="E4:41_15_2_5"/>
<font color="Maroon">rl</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, , <a class="txt" href="jordan12.html#E14:41"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan12.html#E15:41"><i><font color="Green">E61</font></i></a>, , <a class="ref" href="jordan12.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan12.html#T10" target="_self">Th10</a>, , , <a class="ref" href="jordan12.html#T15" target="_self">Th15</a>, , , <a class="ref" href="jordan1g.html#T6">JORDAN1G:6</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E249</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_15_2_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_15_2_6"/><i><font color="Green">E250</font></i>: 
( <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <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">f</font></span>)</span> )
 ;<br/><a NAME="E2:41_15_2_6"/><b>then </b>
 ex <font color="Olive">r</font>, <font color="Olive">s</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">rl'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r</font>,<font color="Olive">s</font></span><a href="euclid.html#K23">]|</a></span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</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">f</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> )
 <b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, , <a class="txt" href="jordan12.html#E14:41"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan12.html#E15:41"><i><font color="Green">E61</font></i></a>, , <a class="ref" href="jordan12.html#T10" target="_self">Th10</a>, , , <a class="ref" href="jordan12.html#T14" target="_self">Th14</a>, , <a class="ref" href="jordan1g.html#T6">JORDAN1G:6</a></i>;<br/><a NAME="E3:41_15_2_6"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</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">f</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">rl'</font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><b>hence </b><a NAME="E4:41_15_2_6"/>
<font color="Maroon">rl</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, , <a class="txt" href="jordan12.html#E14:41"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan12.html#E15:41"><i><font color="Green">E61</font></i></a>, , <a class="ref" href="jordan12.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan12.html#T10" target="_self">Th10</a>, , , , , <a class="ref" href="jordan1g.html#T6">JORDAN1G:6</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E251</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_15_2_7"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_15_2_7"/><i><font color="Green">E252</font></i>: 
( <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 &amp; 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> )
 ;<br/><a NAME="E2:41_15_2_7"/><b>then </b>
 ex <font color="Olive">r</font>, <font color="Olive">s</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">rp'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r</font>,<font color="Olive">s</font></span><a href="euclid.html#K23">]|</a></span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">s</font> <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">f</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="jordan12.html#T5" target="_self">Th5</a>, , <a class="txt" href="jordan12.html#E14:41"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan12.html#E15:41"><i><font color="Green">E61</font></i></a>, , <a class="ref" href="jordan12.html#T10" target="_self">Th10</a>, , , <a class="ref" href="jordan12.html#T16" target="_self">Th16</a>, , <a class="ref" href="jordan1g.html#T6">JORDAN1G:6</a></i>;<br/><a NAME="E3:41_15_2_7"/><b>then </b>
<font color="Maroon">rp'</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">f</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="euclid.html#T56">EUCLID:56</a></i>;<br/><b>hence </b><a NAME="E4:41_15_2_7"/>
<font color="Maroon">rp</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, , <a class="txt" href="jordan12.html#E14:41"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan12.html#E15:41"><i><font color="Green">E61</font></i></a>, , <a class="ref" href="jordan12.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan12.html#T10" target="_self">Th10</a>, , , , , <a class="ref" href="jordan1g.html#T6">JORDAN1G:6</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E253</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_15_2_8"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_15_2_8"/><i><font color="Green">E254</font></i>: 
( <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 &amp; 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j1</font> )
 ;<br/><a NAME="E2:41_15_2_8"/><b>then </b>
 ex <font color="Olive">r</font>, <font color="Olive">s</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">rp'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r</font>,<font color="Olive">s</font></span><a href="euclid.html#K23">]|</a></span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, , <a class="txt" href="jordan12.html#E14:41"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan12.html#E15:41"><i><font color="Green">E61</font></i></a>, , <a class="ref" href="jordan12.html#T10" target="_self">Th10</a>, , , <a class="ref" href="jordan12.html#T16" target="_self">Th16</a>, , <a class="ref" href="jordan1g.html#T6">JORDAN1G:6</a></i>;<br/><a NAME="E3:41_15_2_8"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">rp'</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">rp'</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">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><b>hence </b><a NAME="E4:41_15_2_8"/>
<font color="Maroon">rp</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, , <a class="txt" href="jordan12.html#E14:41"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan12.html#E15:41"><i><font color="Green">E61</font></i></a>, , <a class="ref" href="jordan12.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan12.html#T10" target="_self">Th10</a>, , , , , <a class="ref" href="jordan1g.html#T6">JORDAN1G:6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_15_2_9"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_2_9_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_15_2_9_1"/>
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 )
 </span><b>by </b><i><a class="txt" href="jordan12.html#E4:41"><i><font color="Green">E54</font></i></a>, , </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_2_9_1_1"><b>suppose </b></a><a NAME="E1:41_15_2_9_1_1"/><i><font color="Green">E255</font></i>: 
<font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:41_15_2_9_1_1"/><i><font color="Green">E256</font></i>: 
<font color="Maroon">rl</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_15_2_9_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_2_9_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_15_2_9_1_1_1_1"/>
( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j1</font> or 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> )
 </span><b>by </b><i><a class="txt" href="jordan12.html#E24:41"><i><font color="Green">E69</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_2_9_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:41_15_2_9_1_1_1_1_1"/>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_15_2_9_1_1_1_1_1"/>
<font color="Maroon">rl</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_2_9_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:41_15_2_9_1_1_1_1_2"/>
1 <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_15_2_9_1_1_1_1_2"/>
<font color="Maroon">rl</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E24"><i><font color="Green">Lemma114</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:41_15_2_9_1_1"/><i><font color="Green">E257</font></i>: 
<font color="Maroon">rp</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_15_2_9_1_1_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_2_9_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_15_2_9_1_1_2_1"/>
( <font color="Maroon">j1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</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">f</font></span>)</span> or <font color="Maroon">j1</font> <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">f</font></span>)</span> )
 </span><b>by </b><i><a class="ref" href="jordan12.html#T6" target="_self">Th6</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_2_9_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:41_15_2_9_1_1_2_1_1"/>
<font color="Maroon">j1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</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">f</font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_15_2_9_1_1_2_1_1"/>
<font color="Maroon">rp</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan12.html#T18" target="_self">Th18</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_2_9_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:41_15_2_9_1_1_2_1_2"/>
<font color="Maroon">j1</font> <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">f</font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_15_2_9_1_1_2_1_2"/>
<font color="Maroon">rp</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E4:41_15_2_9_1_1"/><i><font color="Green">E258</font></i>: 
<font color="Maroon">rl</font> <a href="hidden.html#R2">in</a>  <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan12.html#T4" target="_self">Th4</a>, <a class="ref" href="jordan12.html#T19" target="_self">Th19</a></i>;<br/><a NAME="E5:41_15_2_9_1_1"/><i><font color="Green">E259</font></i>: 
<font color="Maroon">rp</font> <a href="hidden.html#R2">in</a>  <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan12.html#T3" target="_self">Th3</a>, </i>;<br/><a NAME="E6:41_15_2_9_1_1"/><i><font color="Green">E260</font></i>: 
not <font color="Maroon">rl</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="jordan12.html#T4" target="_self">Th4</a>, <a class="ref" href="jordan12.html#T19" target="_self">Th19</a></i>;<br/><a NAME="E7:41_15_2_9_1_1"/>
not <font color="Maroon">rp</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="jordan12.html#T3" target="_self">Th3</a>, </i>;<br/><b>hence </b><a NAME="E8:41_15_2_9_1_1"/>
for <font color="Olive">C</font> being  <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>  holds <br/>(  not <font color="Olive">C</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">f</font></span>)</span> <a href="subset_1.html#K3">`</a>  or  not <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> or  not <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 <b>by </b><i><a class="txt" href="jordan12.html#E1"><i><font color="Green">Lemma40</font></i></a>, <a class="txt" href="jordan12.html#E1:41"><i><font color="Green">E42</font></i></a>, , <a class="txt" href="jordan12.html#E3"><i><font color="Green">Lemma49</font></i></a>, , , <a class="ref" href="jordan12.html#T15" target="_self">Th15</a>, <a class="ref" href="jordan12.html#T17" target="_self">Th17</a>, <a class="txt" href="jordan12.html#E26"><i><font color="Green">Lemma123</font></i></a>, <a class="ref" href="jordan12.html#T20" target="_self">Th20</a>, <a class="ref" href="jordan12.html#T21" target="_self">Th21</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_2_9_1_2"><b>suppose </b></a><a NAME="E1:41_15_2_9_1_2"/><i><font color="Green">E261</font></i>: 
<font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:41_15_2_9_1_2"/><i><font color="Green">E262</font></i>: 
<font color="Maroon">rp</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_15_2_9_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_2_9_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_15_2_9_1_2_1_1"/>
( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j1</font> or 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> )
 </span><b>by </b><i><a class="txt" href="jordan12.html#E24:41"><i><font color="Green">E69</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_2_9_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:41_15_2_9_1_2_1_1_1"/>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_15_2_9_1_2_1_1_1"/>
<font color="Maroon">rp</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_2_9_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:41_15_2_9_1_2_1_1_2"/>
1 <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_15_2_9_1_2_1_1_2"/>
<font color="Maroon">rp</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:41_15_2_9_1_2"/><i><font color="Green">E263</font></i>: 
<font color="Maroon">rl</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_15_2_9_1_2_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_2_9_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_15_2_9_1_2_2_1"/>
( <font color="Maroon">j1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</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">f</font></span>)</span> or <font color="Maroon">j1</font> <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">f</font></span>)</span> )
 </span><b>by </b><i><a class="ref" href="jordan12.html#T6" target="_self">Th6</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_2_9_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:41_15_2_9_1_2_2_1_1"/>
<font color="Maroon">j1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</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">f</font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_15_2_9_1_2_2_1_1"/>
<font color="Maroon">rl</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_2_9_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:41_15_2_9_1_2_2_1_2"/>
<font color="Maroon">j1</font> <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">f</font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_15_2_9_1_2_2_1_2"/>
<font color="Maroon">rl</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E4:41_15_2_9_1_2"/><b>then </b><i><font color="Green">E264</font></i>: 
<font color="Maroon">rl</font> <a href="hidden.html#R2">in</a>  <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan12.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E5:41_15_2_9_1_2"/><i><font color="Green">E265</font></i>: 
<font color="Maroon">rp</font> <a href="hidden.html#R2">in</a>  <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan12.html#T3" target="_self">Th3</a>, <a class="ref" href="jordan12.html#T19" target="_self">Th19</a></i>;<br/><a NAME="E6:41_15_2_9_1_2"/><i><font color="Green">E266</font></i>: 
not <font color="Maroon">rl</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="jordan12.html#T4" target="_self">Th4</a>, </i>;<br/><a NAME="E7:41_15_2_9_1_2"/>
not <font color="Maroon">rp</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="jordan12.html#T3" target="_self">Th3</a>, <a class="ref" href="jordan12.html#T19" target="_self">Th19</a></i>;<br/><b>hence </b><a NAME="E8:41_15_2_9_1_2"/>
for <font color="Olive">C</font> being  <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>  holds <br/>(  not <font color="Olive">C</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">f</font></span>)</span> <a href="subset_1.html#K3">`</a>  or  not <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> or  not <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 <b>by </b><i><a class="txt" href="jordan12.html#E1"><i><font color="Green">Lemma40</font></i></a>, <a class="txt" href="jordan12.html#E1:41"><i><font color="Green">E42</font></i></a>, , <a class="txt" href="jordan12.html#E3"><i><font color="Green">Lemma49</font></i></a>, , , <a class="ref" href="jordan12.html#T15" target="_self">Th15</a>, <a class="ref" href="jordan12.html#T17" target="_self">Th17</a>, <a class="txt" href="jordan12.html#E26"><i><font color="Green">Lemma123</font></i></a>, <a class="ref" href="jordan12.html#T20" target="_self">Th20</a>, <a class="ref" href="jordan12.html#T21" target="_self">Th21</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E24:41_15_2"/>
for <font color="Olive">C</font> being  <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>  holds <br/>(  not <font color="Olive">C</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">f</font></span>)</span> <a href="subset_1.html#K3">`</a>  or  not <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> or  not <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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