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

<b>let </b><font color="Maroon">f1</font> be   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">f2</font> be   <a href="struct_0.html#NM7">FinSequence</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="E1:4"/><i><font color="Green">E43</font></i>: 
( 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> &amp; 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> )
 <b>and </b><br/><a NAME="E2:4"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">f1</font> is <a href="topreal1.html#V1">special</a>
 <b>and </b><br/><a NAME="E3:4"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">f2</font> is <a href="topreal1.html#V1">special</a>
 <b>and </b><br/><a NAME="E4:4"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f1</font> &amp; <font color="Olive">n</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">f1</font> holds <br/><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>and </b><br/><a NAME="E5:4"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f2</font> &amp; <font color="Olive">n</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">f2</font> holds <br/><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>and </b><br/><a NAME="E6:4"/><i><font color="Green">E49</font></i>: 
 <a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f1</font> <a href="goboard4.html#R1" target="_self">lies_between</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f1</font></span>)</span> <a href="goboard1.html#K1">.</a> 1,<span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f1</font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span>
 <b>and </b><br/><a NAME="E7:4"/><i><font color="Green">E50</font></i>: 
 <a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f2</font> <a href="goboard4.html#R1" target="_self">lies_between</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f1</font></span>)</span> <a href="goboard1.html#K1">.</a> 1,<span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f1</font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span>
 <b>and </b><br/><a NAME="E8:4"/><i><font color="Green">E51</font></i>: 
 <a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f1</font> <a href="goboard4.html#R1" target="_self">lies_between</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f2</font></span>)</span> <a href="goboard1.html#K1">.</a> 1,<span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f2</font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font></span>)</span>
 <b>and </b><br/><a NAME="E9:4"/><i><font color="Green">E53</font></i>: 
 <a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f2</font> <a href="goboard4.html#R1" target="_self">lies_between</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f2</font></span>)</span> <a href="goboard1.html#K1">.</a> 1,<span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f2</font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font></span>)</span>
 ;<br/>

<a NAME="E10:4"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">f1</font>, <font color="Maroon">f2</font> = <font color="Maroon">f2</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM7">FinSequence</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="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f2</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E13:4"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E14:4"/><b>then </b><i><font color="Green">E54</font></i>: 
2 <a href="nat_1.html#K1">+</a> 2 <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>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<b>set </b><font color="Maroon">G</font> =  <a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:4_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f1</font>
 ;<br/><a NAME="E2:4_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f1</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T39">FINSEQ_1:39</a>, <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>;<br/><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j</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="E4:4_1"/><i><font color="Green">E56</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</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">n</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">i</font>,<font color="Maroon">j</font> )
 <b>by </b><i>, <a class="ref" href="goboard2.html#T25">GOBOARD2:25</a></i>;<br/><b>take </b><font color="Maroon">i</font> = <font color="Maroon">i</font>;<br/><b>take </b><font color="Maroon">j</font> = <font color="Maroon">j</font>;<br/><b>thus </b><a NAME="E5:4_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</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">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</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">i</font>,<font color="Maroon">j</font> )
 <b>by </b><i>, , <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon">g1</font> being   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E17:4"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">g1</font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font> &amp;  <a href="topreal1.html#K5">L~</a> <font color="Maroon">g1</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 1 &amp; <font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> )
 <b>by </b><i>, , <a class="ref" href="goboard2.html#T17">GOBOARD2:17</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:4_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f2</font>
 ;<br/><a NAME="E2:4_2"/><b>then </b>
( <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</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="ref" href="finseq_1.html#T41">FINSEQ_1:41</a>, <a class="ref" href="finseq_4.html#T84">FINSEQ_4:84</a></i>;<br/><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j</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="E4:4_2"/><i><font color="Green">E59</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> )
 <b>by </b><i><a class="ref" href="goboard2.html#T25">GOBOARD2:25</a></i>;<br/><b>take </b><font color="Maroon">i</font> = <font color="Maroon">i</font>;<br/><b>take </b><font color="Maroon">j</font> = <font color="Maroon">j</font>;<br/><b>thus </b><a NAME="E5:4_2"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</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">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</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">i</font>,<font color="Maroon">j</font> )
 <b>by </b><i>, , <a class="ref" href="finseq_4.html#T84">FINSEQ_4:84</a></i>;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon">g2</font> being   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E20:4"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">g2</font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font> &amp;  <a href="topreal1.html#K5">L~</a> <font color="Maroon">g2</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f2</font> &amp; <font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> 1 &amp; <font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font> )
 <b>by </b><i>, , <a class="ref" href="goboard2.html#T17">GOBOARD2:17</a></i>;<br/>
<b>set </b><font color="Maroon">x</font> =  <a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f</font>;<br/>
<b>set </b><font color="Maroon">y</font> =  <a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f</font>;<br/>
<b>set </b><font color="Maroon">x1</font> =  <a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f1</font>;<br/>
<b>set </b><font color="Maroon">y1</font> =  <a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f1</font>;<br/>
<b>set </b><font color="Maroon">x2</font> =  <a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f2</font>;<br/>
<b>set </b><font color="Maroon">y2</font> =  <a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f2</font>;<br/>
<a NAME="E21:4"/><i><font color="Green">E61</font></i>: 
(  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f1</font> &amp;  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f2</font> &amp;  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f</font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> &amp;  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f1</font></span>)</span> &amp;  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f1</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> &amp;  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f2</font></span>)</span> &amp;  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f2</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f1</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f2</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</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="finseq_1.html#T39">FINSEQ_1:39</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a>, <a class="ref" href="goboard1.html#D3">GOBOARD1:def 3</a>, <a class="ref" href="goboard1.html#D4">GOBOARD1:def 4</a></i>;<br/>
<a NAME="E22:4"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> )
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E23:4"/><b>then </b><i><font color="Green">E62</font></i>: 
( 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f1</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f1</font> &amp; 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f2</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f2</font> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E24:4"/><b>then </b><i><font color="Green">E63</font></i>: 
( <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 1 &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> 1 &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span> <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> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font></span>)</span> <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="ref" href="finseq_1.html#T41">FINSEQ_1:41</a>, <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a>, <a class="ref" href="finseq_4.html#T84">FINSEQ_4:84</a></i>;<br/>
<b>reconsider </b><font color="Maroon">p1</font> = <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 1, <font color="Maroon">p2</font> = <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span>, <font color="Maroon">q1</font> = <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> 1, <font color="Maroon">q2</font> = <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font></span>)</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="E26:4"/><i><font color="Green">E66</font></i>: 
( <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f</font></span>)</span> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f1</font></span>)</span> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f</font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f1</font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f</font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f2</font></span>)</span> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f</font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f2</font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> <a href="euclid.html#K22">`2</a>  )
 
<b>by </b><i>, , , <a class="ref" href="goboard1.html#D3">GOBOARD1:def 3</a>, <a class="ref" href="goboard1.html#D4">GOBOARD1:def 4</a></i>;<br/>
<a NAME="E27:4"/><i><font color="Green">E67</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:4_3"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 ;<br/><a NAME="E2:4_3"/><b>then </b><i><font color="Green">E69</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <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><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>set </b><font color="Maroon">s</font> = <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_3_1_1"/>
( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_3_1_1_1"><b>suppose </b></a><a NAME="E1:4_3_1_1_1"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font>
 ;<br/><div class="add"><a NAME="E2:4_3_1_1_1"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f1</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>reconsider </b><font color="Maroon">u</font> = <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</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> ;<br/><a NAME="E4:4_3_1_1_1"/>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>;<br/><a NAME="E5:4_3_1_1_1"/><b>then </b><i><font color="Green">E72</font></i>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, , <a class="ref" href="goboard1.html#D3">GOBOARD1:def 3</a></i>;<br/><a NAME="E6:4_3_1_1_1"/>
<span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, , <a class="ref" href="goboard1.html#D3">GOBOARD1:def 3</a></i>;<br/><b>hence </b><a NAME="E7:4_3_1_1_1"/>
<font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, , , , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_3_1_1_2"><b>suppose </b></a><a NAME="E1:4_3_1_1_2"/><i><font color="Green">E73</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">w5</font> = <font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E3:4_3_1_1_2"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w5</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span>
 ;<br/><a NAME="E4:4_3_1_1_2"/>
<font color="Maroon">w5</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/><a NAME="E5:4_3_1_1_2"/><b>then </b><i><font color="Green">E76</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">w5</font> &amp; <font color="Maroon">w5</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> )
 <b>by </b><i>, , , <a class="ref" href="nat_1.html#T39">NAT_1:39</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E6:4_3_1_1_2"/><b>then </b><i><font color="Green">E77</font></i>: 
( <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">w5</font> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 <b>by </b><i>, , <a class="ref" href="goboard2.html#T5">GOBOARD2:5</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>reconsider </b><font color="Maroon">m1</font> = <font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/><a NAME="E8:4_3_1_1_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">m1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f2</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>reconsider </b><font color="Maroon">u</font> = <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m1</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> ;<br/><a NAME="E10:4_3_1_1_2"/><i><font color="Green">E80</font></i>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, , , <a class="ref" href="goboard1.html#D3">GOBOARD1:def 3</a></i>;<br/><a NAME="E11:4_3_1_1_2"/>
<span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f2</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">m1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, , <a class="ref" href="goboard1.html#D3">GOBOARD1:def 3</a></i>;<br/><b>hence </b><a NAME="E12:4_3_1_1_2"/>
<font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, , , , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:4_3"/>
<font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E29:4"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span>,1</span>)</span>
 
<b>by </b><i>, , <a class="ref" href="goboard2.html#T26">GOBOARD2:26</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:4_4"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 ;<br/><a NAME="E2:4_4"/><b>then </b><i><font color="Green">E83</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <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><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>set </b><font color="Maroon">s</font> = <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_4_1_1"/>
( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_4_1_1_1"><b>suppose </b></a><a NAME="E1:4_4_1_1_1"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font>
 ;<br/><div class="add"><a NAME="E2:4_4_1_1_1"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f1</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>reconsider </b><font color="Maroon">u</font> = <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</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> ;<br/><a NAME="E4:4_4_1_1_1"/>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>;<br/><a NAME="E5:4_4_1_1_1"/><b>then </b><i><font color="Green">E86</font></i>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, , <a class="ref" href="goboard1.html#D3">GOBOARD1:def 3</a></i>;<br/><a NAME="E6:4_4_1_1_1"/>
<span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, , <a class="ref" href="goboard1.html#D3">GOBOARD1:def 3</a></i>;<br/><b>hence </b><a NAME="E7:4_4_1_1_1"/>
<span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, , , , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_4_1_1_2"><b>suppose </b></a><a NAME="E1:4_4_1_1_2"/><i><font color="Green">E87</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">w5</font> = <font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E3:4_4_1_1_2"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w5</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span>
 ;<br/><a NAME="E4:4_4_1_1_2"/>
<font color="Maroon">w5</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/><a NAME="E5:4_4_1_1_2"/><b>then </b><i><font color="Green">E89</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">w5</font> &amp; <font color="Maroon">w5</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> )
 <b>by </b><i>, , , <a class="ref" href="nat_1.html#T39">NAT_1:39</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E6:4_4_1_1_2"/><b>then </b><i><font color="Green">E90</font></i>: 
( <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">w5</font> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 <b>by </b><i>, , <a class="ref" href="goboard2.html#T5">GOBOARD2:5</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>reconsider </b><font color="Maroon">m1</font> = <font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/><a NAME="E8:4_4_1_1_2"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">m1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f2</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>reconsider </b><font color="Maroon">u</font> = <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m1</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> ;<br/><a NAME="E10:4_4_1_1_2"/><i><font color="Green">E93</font></i>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, , , <a class="ref" href="goboard1.html#D3">GOBOARD1:def 3</a></i>;<br/><a NAME="E11:4_4_1_1_2"/>
<span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f2</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">m1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, , <a class="ref" href="goboard1.html#D3">GOBOARD1:def 3</a></i>;<br/><b>hence </b><a NAME="E12:4_4_1_1_2"/>
<span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, , , , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:4_4"/>
<span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> 
 ;<br/></div><b>end;</b></div>
<a NAME="E31:4"/><b>then </b><i><font color="Green">E94</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</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"><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></span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="goboard2.html#T27">GOBOARD2:27</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:4_5"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 ;<br/><a NAME="E2:4_5"/><b>then </b><i><font color="Green">E96</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <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><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>set </b><font color="Maroon">s</font> = <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_5_1_1"/>
( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_5_1_1_1"><b>suppose </b></a><a NAME="E1:4_5_1_1_1"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font>
 ;<br/><div class="add"><a NAME="E2:4_5_1_1_1"/><b>then </b><i><font color="Green">E98</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f1</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>reconsider </b><font color="Maroon">u</font> = <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</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> ;<br/><a NAME="E4:4_5_1_1_1"/>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>;<br/><a NAME="E5:4_5_1_1_1"/><b>then </b><i><font color="Green">E99</font></i>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, , <a class="ref" href="goboard1.html#D4">GOBOARD1:def 4</a></i>;<br/><a NAME="E6:4_5_1_1_1"/>
<span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, , <a class="ref" href="goboard1.html#D4">GOBOARD1:def 4</a></i>;<br/><b>hence </b><a NAME="E7:4_5_1_1_1"/>
<font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, , , , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_5_1_1_2"><b>suppose </b></a><a NAME="E1:4_5_1_1_2"/><i><font color="Green">E100</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">w5</font> = <font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E3:4_5_1_1_2"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w5</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span>
 ;<br/><a NAME="E4:4_5_1_1_2"/>
<font color="Maroon">w5</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/><a NAME="E5:4_5_1_1_2"/><b>then </b><i><font color="Green">E102</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">w5</font> &amp; <font color="Maroon">w5</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> )
 <b>by </b><i>, , , <a class="ref" href="nat_1.html#T39">NAT_1:39</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E6:4_5_1_1_2"/><b>then </b><i><font color="Green">E104</font></i>: 
( <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">w5</font> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 <b>by </b><i>, , <a class="ref" href="goboard2.html#T5">GOBOARD2:5</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>reconsider </b><font color="Maroon">m1</font> = <font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/><a NAME="E8:4_5_1_1_2"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">m1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f2</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>reconsider </b><font color="Maroon">u</font> = <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m1</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> ;<br/><a NAME="E10:4_5_1_1_2"/><i><font color="Green">E111</font></i>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, , , <a class="ref" href="goboard1.html#D4">GOBOARD1:def 4</a></i>;<br/><a NAME="E11:4_5_1_1_2"/>
<span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f2</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">m1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, , <a class="ref" href="goboard1.html#D4">GOBOARD1:def 4</a></i>;<br/><b>hence </b><a NAME="E12:4_5_1_1_2"/>
<font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, , , , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:4_5"/>
<font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E33:4"/><b>then </b><i><font color="Green">E112</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span> <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> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span>,1</span>)</span>
 
<b>by </b><i>, , <a class="ref" href="goboard2.html#T28">GOBOARD2:28</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:4_6"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 ;<br/><a NAME="E2:4_6"/><b>then </b><i><font color="Green">E115</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <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><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>set </b><font color="Maroon">s</font> = <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_6_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_6_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_6_1_1"/>
( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_6_1_1_1"><b>suppose </b></a><a NAME="E1:4_6_1_1_1"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font>
 ;<br/><div class="add"><a NAME="E2:4_6_1_1_1"/><b>then </b><i><font color="Green">E116</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f1</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>reconsider </b><font color="Maroon">u</font> = <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</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> ;<br/><a NAME="E4:4_6_1_1_1"/>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>;<br/><a NAME="E5:4_6_1_1_1"/><b>then </b><i><font color="Green">E118</font></i>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, , <a class="ref" href="goboard1.html#D4">GOBOARD1:def 4</a></i>;<br/><a NAME="E6:4_6_1_1_1"/>
<span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, , <a class="ref" href="goboard1.html#D4">GOBOARD1:def 4</a></i>;<br/><b>hence </b><a NAME="E7:4_6_1_1_1"/>
<span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q2</font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, , , , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_6_1_1_2"><b>suppose </b></a><a NAME="E1:4_6_1_1_2"/><i><font color="Green">E119</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">w5</font> = <font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E3:4_6_1_1_2"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w5</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span>
 ;<br/><a NAME="E4:4_6_1_1_2"/>
<font color="Maroon">w5</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/><a NAME="E5:4_6_1_1_2"/><b>then </b><i><font color="Green">E121</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">w5</font> &amp; <font color="Maroon">w5</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> )
 <b>by </b><i>, , , <a class="ref" href="nat_1.html#T39">NAT_1:39</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E6:4_6_1_1_2"/><b>then </b><i><font color="Green">E122</font></i>: 
( <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">w5</font> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 <b>by </b><i>, , <a class="ref" href="goboard2.html#T5">GOBOARD2:5</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>reconsider </b><font color="Maroon">m1</font> = <font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/><a NAME="E8:4_6_1_1_2"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">m1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f2</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>reconsider </b><font color="Maroon">u</font> = <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m1</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> ;<br/><a NAME="E10:4_6_1_1_2"/><i><font color="Green">E124</font></i>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, , , <a class="ref" href="goboard1.html#D4">GOBOARD1:def 4</a></i>;<br/><a NAME="E11:4_6_1_1_2"/>
<span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f2</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">m1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, , <a class="ref" href="goboard1.html#D4">GOBOARD1:def 4</a></i>;<br/><b>hence </b><a NAME="E12:4_6_1_1_2"/>
<span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q2</font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, , , , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:4_6"/>
<span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q2</font> <a href="euclid.html#K22">`2</a> 
 ;<br/></div><b>end;</b></div>
<a NAME="E35:4"/><b>then </b><i><font color="Green">E125</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</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"><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>
 
<b>by </b><i>, , <a class="ref" href="goboard2.html#T29">GOBOARD2:29</a></i>;<br/>
<a NAME="E36:4"/>
( 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> &amp; 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font> )
 
<b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E37:4"/>
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f2</font>
 <b>by </b><i>, , , , , , , </i>;<br/>


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