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<div class="add">

<b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="compts_1.html#V6">compact</a> non <a href="sppol_1.html#V1">horizontal</a> non <a href="sppol_1.html#V2">vertical</a> <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>;<br/>

<b>set </b><font color="Maroon">f</font> =  <a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font>;<br/>
<b>assume </b><a NAME="E1:104"/>
 <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> is <a href="jordan2c.html#V1" target="_self">Bounded</a>
 ;<br/>

<b>then consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:104"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">q</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive">q</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> holds <br/><span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">q3</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">q4</font> = <font color="Maroon">q3</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="E6:104"/><i><font color="Green">E56</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q4</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i/>;<br/>
<a NAME="E7:104"/><i><font color="Green">E57</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q4</font></span><a href="toprns_1.html#K5">.|</a></span>
 
<b>by </b><i><a class="ref" href="toprns_1.html#T26">TOPRNS_1:26</a></i>;<br/>
<b>set </b><font color="Maroon">r1</font> = <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>;<br/>
<a NAME="E8:104"/><i><font color="Green">E58</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="toprns_1.html#T26">TOPRNS_1:26</a></i>;<br/>
<a NAME="E9:104"/><i><font color="Green">E59</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="binop_2.html#K9">+</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T17" target="_self">Th17</a>, <a class="ref" href="jordan2c.html#D4" target="_self">Def4</a>, <a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<a NAME="E10:104"/><i><font color="Green">E60</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K17">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="binop_2.html#K9">+</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K9">+</a> 1
 
<b>by </b><i><a class="ref" href="jordan2c.html#T18" target="_self">Th18</a>, , <a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<b>set </b><font color="Maroon">q1</font> = <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="binop_2.html#K9">+</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E11:104"/><i><font color="Green">E61</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="binop_2.html#K9">+</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K17">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="binop_2.html#K9">+</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span></span><a href="toprns_1.html#K5">.|</a></span> <a href="binop_2.html#K10">-</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
 
<b>by </b><i><a class="ref" href="toprns_1.html#T32">TOPRNS_1:32</a></i>;<br/>
<b>set </b><font color="Maroon">p</font> = <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="binop_2.html#K9">+</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>;<br/>
<a NAME="E12:104"/><i><font color="Green">E63</font></i>: 
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="binop_2.html#K9">+</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0 &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="binop_2.html#K9">+</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K17">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="binop_2.html#K9">+</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K9">+</a> 1 )
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<i><font color="Green">E64</font></i>: <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="binop_2.html#K9">+</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span></span><a href="toprns_1.html#K5">.|</a></span> = 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="binop_2.html#K9">+</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="binop_2.html#K9">+</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="jgraph_1.html#T47">JGRAPH_1:47</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K17">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="binop_2.html#K9">+</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K9">+</a> 1
<b>by </b><i>, , <a class="ref" href="square_1.html#T89">SQUARE_1:89</a></i>
;<br/>
<a NAME="E14:104"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="binop_2.html#K9">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<a NAME="E15:104"/><b>then </b><i><font color="Green">E65</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="binop_2.html#K9">+</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K17">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T21" target="_self">Th21</a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f1</font> =  <a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font> as  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/>
<a NAME="E17:104"/><i><font color="Green">E66</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> 5
 
<b>by </b><i><a class="ref" href="sprect_1.html#T90">SPRECT_1:90</a></i>;<br/>
<a NAME="E18:104"/><b>then </b><i><font color="Green">E67</font></i>: 
1 <a href="hidden.html#R2">in</a>  <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>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E19:104"/><b>then </b><i><font color="Green">E130</font></i>: 
1 <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_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E20:104"/>
2 <a href="hidden.html#R2">in</a>  <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>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T23" target="_self">Th23</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E21:104"/><b>then </b><i><font color="Green">E131</font></i>: 
2 <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_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E22:104"/>
3 <a href="hidden.html#R2">in</a>  <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>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T23" target="_self">Th23</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E23:104"/><b>then </b><i><font color="Green">E152</font></i>: 
3 <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_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E24:104"/>
4 <a href="hidden.html#R2">in</a>  <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>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T23" target="_self">Th23</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E25:104"/><b>then </b><i><font color="Green">E154</font></i>: 
4 <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_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E26:104"/>
5 <a href="hidden.html#R2">in</a>  <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>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T23" target="_self">Th23</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E27:104"/><b>then </b><i><font color="Green">E157</font></i>: 
5 <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_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E28:104"/><i><font color="Green">E158</font></i>: 
 <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="sprect_1.html#T33">SPRECT_1:33</a></i>;<br/>
<a NAME="E29:104"/><i><font color="Green">E159</font></i>: 
 <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="pscomp_1.html#K16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="sprect_1.html#T34">SPRECT_1:34</a></i>;<br/>
<a NAME="E30:104"/><i><font color="Green">E160</font></i>: 
( <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K29">N-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K28">W-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="sprect_1.html#T91">SPRECT_1:91</a></i>;<br/>
<a NAME="E31:104"/><i><font color="Green">E161</font></i>: 
( <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K30">N-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="sprect_1.html#T92">SPRECT_1:92</a></i>;<br/>
<a NAME="E32:104"/><i><font color="Green">E162</font></i>: 
( <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 3 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K33">S-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 3 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K32">E-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="sprect_1.html#T93">SPRECT_1:93</a></i>;<br/>
<a NAME="E33:104"/><i><font color="Green">E163</font></i>: 
( <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 4 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K34">S-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 4 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="sprect_1.html#T94">SPRECT_1:94</a></i>;<br/>
<a NAME="E34:104"/><i><font color="Green">E164</font></i>: 
<font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 5 <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 1
 
<b>by </b><i><a class="ref" href="jordan2c.html#T23" target="_self">Th23</a>, <a class="ref" href="finseq_6.html#D1">FINSEQ_6:def 1</a></i>;<br/>
<a NAME="E35:104"/><i><font color="Green">E165</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 1</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">f1</font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i><a class="ref" href="jordan2c.html#T29" target="_self">Th29</a>, <a class="ref" href="jordan2c.html#T30" target="_self">Th30</a>, <a class="ref" href="sprect_2.html#T55">SPRECT_2:55</a></i>;<br/>
<a NAME="E36:104"/><i><font color="Green">E166</font></i>: 
 <a href="goboard2.html#K3">GoB</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a>  <a href="goboard2.html#K1">GoB</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2">Incr</a> <span class="p2">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f1</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="goboard2.html#K2">Incr</a> <span class="p2">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="goboard2.html#D3">GOBOARD2:def 3</a></i>;<br/>
<a NAME="E37:104"/><i><font color="Green">E167</font></i>: 
<font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K28">W-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="sprect_1.html#T91">SPRECT_1:91</a></i>;<br/>
<a NAME="E38:104"/>
1 <a href="hidden.html#R2">in</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>
 
<b>by </b><i>, <a class="ref" href="sprect_2.html#T19">SPRECT_2:19</a></i>;<br/>
<a NAME="E39:104"/><b>then </b>
<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> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <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="E40:104"/><b>then </b><i><font color="Green">E168</font></i>: 
<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>  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T37" target="_self">Th37</a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E41:104"/><i><font color="Green">E169</font></i>: 
<font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 2 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="sprect_1.html#T92">SPRECT_1:92</a></i>;<br/>
<a NAME="E42:104"/><i><font color="Green">E170</font></i>: 
2 <a href="hidden.html#R2">in</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>
 
<b>by </b><i>, <a class="ref" href="sprect_2.html#T19">SPRECT_2:19</a></i>;<br/>
<a NAME="E43:104"/><b>then </b>
<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> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <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="E44:104"/><b>then </b><i><font color="Green">E171</font></i>: 
<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> 2 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T39" target="_self">Th39</a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E45:104"/><i><font color="Green">E172</font></i>: 
<font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 3 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K32">E-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="sprect_1.html#T93">SPRECT_1:93</a></i>;<br/>
<a NAME="E46:104"/>
3 <a href="hidden.html#R2">in</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>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T25" target="_self">Th25</a>, <a class="ref" href="sprect_2.html#T19">SPRECT_2:19</a></i>;<br/>
<a NAME="E47:104"/><b>then </b>
<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> 3 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <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="E48:104"/><b>then </b><i><font color="Green">E205</font></i>: 
<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> 3 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E49:104"/><i><font color="Green">E206</font></i>: 
<font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 4 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="sprect_1.html#T94">SPRECT_1:94</a></i>;<br/>
<a NAME="E50:104"/>
4 <a href="hidden.html#R2">in</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>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T26" target="_self">Th26</a>, <a class="ref" href="sprect_2.html#T19">SPRECT_2:19</a></i>;<br/>
<a NAME="E51:104"/><b>then </b>
<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> 4 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <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="E52:104"/><b>then </b><i><font color="Green">E207</font></i>: 
<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> 4 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E53:104"/><i><font color="Green">E208</font></i>: 
<font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 5 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K28">W-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T23" target="_self">Th23</a>, <a class="ref" href="jordan2c.html#T29" target="_self">Th29</a>, <a class="ref" href="finseq_6.html#D1">FINSEQ_6:def 1</a></i>;<br/>
<a NAME="E54:104"/>
5 <a href="hidden.html#R2">in</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>
 
<b>by </b><i>, <a class="ref" href="sprect_2.html#T19">SPRECT_2:19</a></i>;<br/>
<a NAME="E55:104"/><b>then </b>
<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> 5 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <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="E56:104"/><b>then </b><i><font color="Green">E209</font></i>: 
<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> 5 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E57:104"/><i><font color="Green">E210</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f1</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:104_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f1</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">u</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:104_2"/><i><font color="Green">E211</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</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; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:104_2"/>
<font color="Maroon">u</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="jordan2c.html#T41" target="_self">Th41</a>, <a class="ref" href="sprect_2.html#T19">SPRECT_2:19</a></i>;<br/>
<a NAME="E5:104_2"/><b>then </b><i><font color="Green">E212</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <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>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">mu</font> = <font color="Maroon">u</font> 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="jordan2c.html#T41" target="_self">Th41</a></i>;<br/>
<a NAME="E7:104_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">mu</font> &amp; <font color="Maroon">mu</font> <a href="xxreal_0.html#R1">&lt;=</a> 5 )
 
<b>by </b><i><a class="ref" href="jordan2c.html#T23" target="_self">Th23</a>, <a class="ref" href="jordan2c.html#T42" target="_self">Th42</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E8:104_2"/><b>then </b><i><font color="Green">E213</font></i>: 
( <font color="Maroon">mu</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">mu</font> <a href="hidden.html#R1">=</a> 1 <a href="binop_2.html#K23">+</a> 1 or <font color="Maroon">mu</font> <a href="hidden.html#R1">=</a> 1 <a href="binop_2.html#K23">+</a> 2 or <font color="Maroon">mu</font> <a href="hidden.html#R1">=</a> 1 <a href="binop_2.html#K23">+</a> 3 or <font color="Maroon">mu</font> <a href="hidden.html#R1">=</a> 1 <a href="binop_2.html#K23">+</a> 4 )
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:104_2_1"/>
( <font color="Maroon">mu</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">mu</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">mu</font> <a href="hidden.html#R1">=</a> 3 or <font color="Maroon">mu</font> <a href="hidden.html#R1">=</a> 4 or <font color="Maroon">mu</font> <a href="hidden.html#R1">=</a> 5 )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_2_1_1"><b>suppose </b></a><a NAME="E1:104_2_1_1"/>
<font color="Maroon">mu</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:104_2_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="ref" href="jordan2c.html#T38" target="_self">Th38</a>, <a class="ref" href="jordan2c.html#T41" target="_self">Th41</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_2_1_2"><b>suppose </b></a><a NAME="E1:104_2_1_2"/>
<font color="Maroon">mu</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><b>hence </b><a NAME="E2:104_2_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i>, <a class="ref" href="jordan2c.html#T41" target="_self">Th41</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_2_1_3"><b>suppose </b></a><a NAME="E1:104_2_1_3"/>
<font color="Maroon">mu</font> <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><b>hence </b><a NAME="E2:104_2_1_3"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="ref" href="jordan2c.html#T40" target="_self">Th40</a>, <a class="ref" href="jordan2c.html#T41" target="_self">Th41</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_2_1_4"><b>suppose </b></a><a NAME="E1:104_2_1_4"/>
<font color="Maroon">mu</font> <a href="hidden.html#R1">=</a> 4
 ;<br/><div class="add"><b>hence </b><a NAME="E2:104_2_1_4"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i>, <a class="ref" href="jordan2c.html#T41" target="_self">Th41</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_2_1_5"><b>suppose </b></a><a NAME="E1:104_2_1_5"/>
<font color="Maroon">mu</font> <a href="hidden.html#R1">=</a> 5
 ;<br/><div class="add"><b>hence </b><a NAME="E2:104_2_1_5"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E19:104"><i><font color="Green">E130</font></i></a>, <a class="ref" href="jordan2c.html#T41" target="_self">Th41</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E58:104"/>
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span></span>)</span></span><a href="domain_1.html#K7">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f1</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:104_3"/><i><font color="Green">E214</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:104_3_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span> or <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span> )
 </span><b>by </b><i><a class="ref" href="jordan2c.html#T41" target="_self">Th41</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_3_1_1"><b>suppose </b></a><a NAME="E1:104_3_1_1"/><i><font color="Green">E215</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:104_3_1_1"/>
1 <a href="hidden.html#R2">in</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>
 <b>by </b><i>, <a class="ref" href="sprect_2.html#T19">SPRECT_2:19</a></i>;<br/><b>hence </b><a NAME="E3:104_3_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f1</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan2c.html#T38" target="_self">Th38</a>, <a class="ref" href="jordan2c.html#T42" target="_self">Th42</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_3_1_2"><b>suppose </b></a><a NAME="E1:104_3_1_2"/><i><font color="Green">E216</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:104_3_1_2"/>
2 <a href="hidden.html#R2">in</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>
 <b>by </b><i>, <a class="ref" href="sprect_2.html#T19">SPRECT_2:19</a></i>;<br/><b>hence </b><a NAME="E3:104_3_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f1</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan2c.html#T42" target="_self">Th42</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E59:104"/><b>then </b><i><font color="Green">E217</font></i>: 
 <a href="relset_1.html#K5">rng</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> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E21:104"><i><font color="Green">E131</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E60:104"/>
3 <a href="hidden.html#R2">in</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>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T25" target="_self">Th25</a>, <a class="ref" href="sprect_2.html#T20">SPRECT_2:20</a></i>;<br/>
<a NAME="E61:104"/><b>then </b>
<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> 3 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <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="E62:104"/><b>then </b><i><font color="Green">E218</font></i>: 
<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> 3 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T31" target="_self">Th31</a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E63:104"/>
2 <a href="hidden.html#R2">in</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>
 
<b>by </b><i>, <a class="ref" href="sprect_2.html#T20">SPRECT_2:20</a></i>;<br/>
<a NAME="E64:104"/><b>then </b>
<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> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <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="E65:104"/><b>then </b><i><font color="Green">E219</font></i>: 
<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> 2 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T30" target="_self">Th30</a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E66:104"/>
1 <a href="hidden.html#R2">in</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>
 
<b>by </b><i>, <a class="ref" href="sprect_2.html#T20">SPRECT_2:20</a></i>;<br/>
<a NAME="E67:104"/><b>then </b>
<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> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <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="E68:104"/><b>then </b><i><font color="Green">E220</font></i>: 
<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> 1 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T29" target="_self">Th29</a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E69:104"/>
4 <a href="hidden.html#R2">in</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>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T26" target="_self">Th26</a>, <a class="ref" href="sprect_2.html#T20">SPRECT_2:20</a></i>;<br/>
<a NAME="E70:104"/><b>then </b>
<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> 4 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <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="E71:104"/><b>then </b><i><font color="Green">E221</font></i>: 
<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> 4 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E72:104"/>
5 <a href="hidden.html#R2">in</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>
 
<b>by </b><i>, <a class="ref" href="sprect_2.html#T20">SPRECT_2:20</a></i>;<br/>
<a NAME="E73:104"/><b>then </b>
<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> 5 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <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="E74:104"/><b>then </b><i><font color="Green">E222</font></i>: 
<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> 5 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T29" target="_self">Th29</a>, <a class="ref" href="jordan2c.html#T33" target="_self">Th33</a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E75:104"/><i><font color="Green">E223</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f1</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_4"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:104_4"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f1</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">u</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:104_4"/><i><font color="Green">E224</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</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; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:104_4"/>
<font color="Maroon">u</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="sprect_2.html#T20">SPRECT_2:20</a></i>;<br/>
<a NAME="E5:104_4"/><b>then </b><i><font color="Green">E225</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <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>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">mu</font> = <font color="Maroon">u</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E7:104_4"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">mu</font> &amp; <font color="Maroon">mu</font> <a href="xxreal_0.html#R1">&lt;=</a> 5 )
 
<b>by </b><i><a class="ref" href="jordan2c.html#T23" target="_self">Th23</a>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E8:104_4"/><b>then </b><i><font color="Green">E232</font></i>: 
( <font color="Maroon">mu</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">mu</font> <a href="hidden.html#R1">=</a> 1 <a href="binop_2.html#K23">+</a> 1 or <font color="Maroon">mu</font> <a href="hidden.html#R1">=</a> 1 <a href="binop_2.html#K23">+</a> 2 or <font color="Maroon">mu</font> <a href="hidden.html#R1">=</a> 1 <a href="binop_2.html#K23">+</a> 3 or <font color="Maroon">mu</font> <a href="hidden.html#R1">=</a> 1 <a href="binop_2.html#K23">+</a> 4 )
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:104_4_1"/>
( <font color="Maroon">mu</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">mu</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">mu</font> <a href="hidden.html#R1">=</a> 3 or <font color="Maroon">mu</font> <a href="hidden.html#R1">=</a> 4 or <font color="Maroon">mu</font> <a href="hidden.html#R1">=</a> 5 )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_4_1_1"><b>suppose </b></a><a NAME="E1:104_4_1_1"/>
<font color="Maroon">mu</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:104_4_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="ref" href="jordan2c.html#T43" target="_self">Th43</a>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_4_1_2"><b>suppose </b></a><a NAME="E1:104_4_1_2"/>
<font color="Maroon">mu</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><b>hence </b><a NAME="E2:104_4_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_4_1_3"><b>suppose </b></a><a NAME="E1:104_4_1_3"/>
<font color="Maroon">mu</font> <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><b>hence </b><a NAME="E2:104_4_1_3"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="ref" href="jordan2c.html#T42" target="_self">Th42</a>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_4_1_4"><b>suppose </b></a><a NAME="E1:104_4_1_4"/>
<font color="Maroon">mu</font> <a href="hidden.html#R1">=</a> 4
 ;<br/><div class="add"><b>hence </b><a NAME="E2:104_4_1_4"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_4_1_5"><b>suppose </b></a><a NAME="E1:104_4_1_5"/>
<font color="Maroon">mu</font> <a href="hidden.html#R1">=</a> 5
 ;<br/><div class="add"><b>hence </b><a NAME="E2:104_4_1_5"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E76:104"/>
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span></span>)</span></span><a href="domain_1.html#K7">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f1</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_5"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:104_5"/><i><font color="Green">E322</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:104_5_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span> or <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span> )
 </span><b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_5_1_1"><b>suppose </b></a><a NAME="E1:104_5_1_1"/><i><font color="Green">E323</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:104_5_1_1"/>
4 <a href="hidden.html#R2">in</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>
 <b>by </b><i><a class="ref" href="jordan2c.html#T26" target="_self">Th26</a>, <a class="ref" href="sprect_2.html#T20">SPRECT_2:20</a></i>;<br/><b>hence </b><a NAME="E3:104_5_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f1</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_5_1_2"><b>suppose </b></a><a NAME="E1:104_5_1_2"/><i><font color="Green">E324</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:104_5_1_2"/>
2 <a href="hidden.html#R2">in</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>
 <b>by </b><i>, <a class="ref" href="sprect_2.html#T20">SPRECT_2:20</a></i>;<br/><b>hence </b><a NAME="E3:104_5_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f1</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E77:104"/><b>then </b><i><font color="Green">E325</font></i>: 
 <a href="relset_1.html#K5">rng</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> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E78:104"/><i><font color="Green">E326</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</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> 2
 
<b>by </b><i><a class="ref" href="jordan2c.html#T27" target="_self">Th27</a>, <a class="ref" href="jordan2c.html#T41" target="_self">Th41</a>, <a class="ref" href="card_2.html#T76">CARD_2:76</a></i>;<br/>
<i><font color="Green">E327</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f1</font></span>)</span> = 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="goboard2.html#T24">GOBOARD2:24</a></i>
<br/>.= 
1 <a href="binop_2.html#K23">+</a> 1
<b>by </b><i><a class="ref" href="jordan2c.html#T27" target="_self">Th27</a>, <a class="ref" href="jordan2c.html#T41" target="_self">Th41</a>, <a class="ref" href="card_2.html#T76">CARD_2:76</a></i>
;<br/>
<i><font color="Green">E328</font></i>:  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f1</font></span>)</span> = 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="goboard2.html#T24">GOBOARD2:24</a></i>
<br/>.= 
1 <a href="binop_2.html#K23">+</a> 1
<b>by </b><i><a class="ref" href="jordan2c.html#T28" target="_self">Th28</a>, , <a class="ref" href="card_2.html#T76">CARD_2:76</a></i>
;<br/>
<a NAME="E81:104"/><i><font color="Green">E329</font></i>: 
1 <a href="hidden.html#R2">in</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="goboard2.html#K3">GoB</a> <font color="Maroon">f1</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E82:104"/><i><font color="Green">E330</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f1</font></span>)</span> <a href="hidden.html#R2">in</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="goboard2.html#K3">GoB</a> <font color="Maroon">f1</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E83:104"/>
for <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of 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>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f1</font></span>)</span> holds <br/> <a href="finseq_1.html#K3">len</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> 2
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be    <a href="finseq_1.html#M2">FinSequence</a> of 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>;<br/>

<b>assume </b><a NAME="E1:104_8"/><i><font color="Green">E331</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f1</font></span>)</span>
 ;<br/>

 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K1">GoB</a> <span class="p2">(<span class="default"><a href="goboard2.html#K2">Incr</a> <span class="p3">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="goboard2.html#K2">Incr</a> <span class="p3">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f1</font></span>)</span></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2">Incr</a> <span class="p2">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="goboard2.html#D1">GOBOARD2:def 1</a></i>
<br/>.= 
2
<b>by </b><i>, <a class="ref" href="goboard2.html#D2">GOBOARD2:def 2</a></i>
;<br/>
<b>then consider </b><font color="Maroon">s1</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E4:104_8"/><i><font color="Green">E332</font></i>: 
( <font color="Maroon">s1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="goboard2.html#K1">GoB</a> <span class="p2">(<span class="default"><a href="goboard2.html#K2">Incr</a> <span class="p3">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="goboard2.html#K2">Incr</a> <span class="p3">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f1</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s1</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#K1">GoB</a> <span class="p2">(<span class="default"><a href="goboard2.html#K2">Incr</a> <span class="p3">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="goboard2.html#K2">Incr</a> <span class="p3">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f1</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="matrix_1.html#D4">MATRIX_1:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">n</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:104_8"/><i><font color="Green">E333</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f1</font></span>)</span> holds <br/> ex <font color="Olive">s</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#D1">MATRIX_1:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">s</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E8:104_8"/><i><font color="Green">E334</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="jordan2c.html#T36" target="_self">Th36</a>, , </i>;<br/>
<b>consider </b><font color="Maroon">s2</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E10:104_8"/><i><font color="Green">E335</font></i>: 
( <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 <b>by </b><i>, </i>;<br/>
<b>thus </b><a NAME="E11:104_8"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 2
 <b>by </b><i>, , , , <a class="ref" href="goboard2.html#D3">GOBOARD2:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E84:104"/><b>then </b><i><font color="Green">E336</font></i>: 
 <a href="goboard2.html#K3">GoB</a> <font color="Maroon">f1</font> is    <a href="matrix_1.html#M1">Matrix</a> of 2,2,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="matrix_1.html#D3">MATRIX_1:def 3</a></i>;<br/>
<a NAME="E85:104"/><i><font color="Green">E337</font></i>: 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f1</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f1</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E86:104"/><b>then </b><i><font color="Green">E338</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default">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">f1</font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f1</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p4">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f1</font></span>)</span></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E87:104"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><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">f1</font></span>)</span></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">f1</font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f1</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p4">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f1</font></span>)</span></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E88:104"/><b>then </b><i><font color="Green">E339</font></i>: 
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">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">f1</font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</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">f1</font></span>)</span> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><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">f1</font></span>)</span></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">f1</font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</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">f1</font></span>)</span> )
 
<b>by </b><i>, , , , <a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>;<br/>
<a NAME="E89:104"/><i><font color="Green">E340</font></i>: 
<font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f1</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f1</font></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E90:104"/><i><font color="Green">E341</font></i>: 
<font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23">+</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">f1</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f1</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f1</font></span>)</span></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E91:104"/>
 <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>
 
<b>by </b><i><a class="ref" href="goboard1.html#D3">GOBOARD1:def 3</a></i>;<br/>
<a NAME="E92:104"/><b>then </b><i><font color="Green">E344</font></i>: 
 <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> <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> <font color="Maroon">f1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E93:104"/><b>then </b><i><font color="Green">E345</font></i>: 
<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> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i><a class="ref" href="jordan2c.html#T24" target="_self">Th24</a>, <a class="ref" href="goboard1.html#D3">GOBOARD1:def 3</a></i>;<br/>
<a NAME="E94:104"/>
<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> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <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="E95:104"/><b>then </b><i><font color="Green">E346</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f1</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f1</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="jordan2c.html#T24" target="_self">Th24</a>, , , <a class="txt" href="jordan2c.html#E23:104"><i><font color="Green">E152</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E96:104"/>
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="domain_1.html#K7">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f1</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_9"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:104_9"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="domain_1.html#K7">}</a></span>
 ;<br/>

<b>hence </b><a NAME="E2:104_9"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f1</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E97:104"/><b>then </b>
<span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="seq_4.html#K1">}</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="seq_4.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="enumset1.html#T41">ENUMSET1:41</a></i>;<br/>
<a NAME="E98:104"/><b>then </b><i><font color="Green">E347</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2"><a href="seq_4.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="seq_4.html#K1">}</a></span> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="seq_4.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="seq_4.html#K1">}</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="card_1.html#T80">CARD_1:80</a></i>;<br/>
<a NAME="E99:104"/>
not <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="seq_4.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T35" target="_self">Th35</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>then </b> <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2"><a href="seq_4.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="seq_4.html#K1">}</a></span> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="seq_4.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="seq_4.html#K1">}</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2"><a href="seq_4.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="seq_4.html#K1">}</a></span></span>)</span> <a href="binop_2.html#K23">+</a> 1
<b>by </b><i><a class="ref" href="card_2.html#T54">CARD_2:54</a></i>
<br/>.= 
1 <a href="binop_2.html#K23">+</a> 1
<b>by </b><i><a class="ref" href="card_1.html#T79">CARD_1:79</a></i>
<br/>.= 
2

;<br/>
<a NAME="E101:104"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</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="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="jordan2c.html#E25:104"><i><font color="Green">E154</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E102:104"/><b>then </b><i><font color="Green">E348</font></i>: 
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">f1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="goboard2.html#T24">GOBOARD2:24</a></i>;<br/>
<a NAME="E103:104"/><b>then </b><i><font color="Green">E349</font></i>: 
1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f1</font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<a NAME="E104:104"/>
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f1</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f1</font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> 1
 
;<br/>
<a NAME="E105:104"/><b>then </b>
 <a href="goboard5.html#K5">left_cell</a> <font color="Maroon">f1</font>,1 <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f1</font></span>)</span>,1,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T23" target="_self">Th23</a>, , , , , <a class="ref" href="goboard5.html#D7">GOBOARD5:def 7</a></i>;<br/>
<a NAME="E106:104"/><b>then </b><i><font color="Green">E350</font></i>: 
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K5">left_cell</a> <font color="Maroon">f1</font>,1</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">r2</font>,<font color="Olive">s</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">r2</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">f1</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r2</font> &amp; <font color="Olive">r2</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">f1</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K23">+</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">f1</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">f1</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="ref" href="goboard6.html#T28">GOBOARD6:28</a></i>;<br/>
<a NAME="E107:104"/><i><font color="Green">E351</font></i>: 
<span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K28">W-max</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T29" target="_self">Th29</a>, <a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/>
<a NAME="E108:104"/>
<span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2 <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K30">N-max</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T30" target="_self">Th30</a>, <a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/>
<a NAME="E109:104"/><b>then </b><i><font color="Green">E352</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K28">W-max</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K30">N-max</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E28:104"><i><font color="Green">E158</font></i></a>, <a class="ref" href="euclid.html#T60">EUCLID:60</a></i>;<br/>
<a NAME="E110:104"/><i><font color="Green">E353</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K30">N-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> = 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
<br/>.= 
2 <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span>

;<br/>
<a NAME="E112:104"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K30">N-max</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E113:104"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K28">W-max</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="binop_2.html#K9">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E29:104"><i><font color="Green">E159</font></i></a>, <a class="ref" href="euclid.html#T62">EUCLID:62</a></i>;<br/>
<b>then </b><i><font color="Green">E354</font></i>: <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="binop_2.html#K9">+</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span> = 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default">0 <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K28">W-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="binop_2.html#K9">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="binop_2.html#K9">+</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T60">EUCLID:60</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K28">W-max</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="binop_2.html#K9">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="binop_2.html#K9">+</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>

;<br/>
<a NAME="E115:104"/><i><font color="Green">E355</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f1</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K28">W-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i><a class="ref" href="jordan2c.html#T23" target="_self">Th23</a>, <a class="ref" href="jordan2c.html#T37" target="_self">Th37</a>, , <a class="ref" href="jordan5d.html#T7">JORDAN5D:7</a></i>;<br/>
<a NAME="E116:104"/><i><font color="Green">E356</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f1</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i><a class="ref" href="jordan2c.html#T23" target="_self">Th23</a>, <a class="ref" href="jordan2c.html#T39" target="_self">Th39</a>, , , <a class="ref" href="jordan5d.html#T7">JORDAN5D:7</a></i>;<br/>
<a NAME="E117:104"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K28">W-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E118:104"/><b>then </b><i><font color="Green">E357</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K28">W-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span></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"><a href="pscomp_1.html#K31">E-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i><a class="ref" href="jordan2c.html#T27" target="_self">Th27</a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E119:104"/><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">f1</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,1</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"><a href="pscomp_1.html#K31">E-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i><a class="txt" href="jordan2c.html#E32:104"><i><font color="Green">E162</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E120:104"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f1</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f1</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K28">W-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E32:104"><i><font color="Green">E162</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E121:104"/><b>then </b><i><font color="Green">E358</font></i>: 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f1</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K28">W-max</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E122:104"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K28">W-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span></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">f1</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i><a class="txt" href="jordan2c.html#E33:104"><i><font color="Green">E163</font></i></a>, <a class="txt" href="jordan2c.html#E34:104"><i><font color="Green">E164</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E123:104"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K28">W-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f1</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f1</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E33:104"><i><font color="Green">E163</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E124:104"/><b>then </b><i><font color="Green">E359</font></i>: 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K28">W-max</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f1</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p4">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E125:104"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f1</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">f1</font></span>)</span></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"><a href="pscomp_1.html#K16">N-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K17">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="binop_2.html#K9">+</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="jordan2c.html#T29" target="_self">Th29</a>, , <a class="txt" href="jordan2c.html#E30:104"><i><font color="Green">E160</font></i></a>, <a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<a NAME="E126:104"/><b>then </b><i><font color="Green">E360</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="binop_2.html#K9">+</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K5">left_cell</a> <font color="Maroon">f1</font>,1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E27:104"><i><font color="Green">E157</font></i></a>, <a class="txt" href="jordan2c.html#E31:104"><i><font color="Green">E161</font></i></a>, <a class="txt" href="jordan2c.html#E35:104"><i><font color="Green">E165</font></i></a>, <a class="txt" href="jordan2c.html#E36:104"><i><font color="Green">E166</font></i></a></i>;<br/>
<a NAME="E127:104"/>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K5">left_cell</a> <font color="Maroon">f1</font>,1</span>)</span> <a href="tarski.html#R1">c=</a>  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span>
 
<b>by </b><i><a class="ref" href="goboard9.html#D1">GOBOARD9:def 1</a></i>;<br/>
<b>hence </b><a NAME="E128:104"/>
contradiction
 <b>by </b><i>, <a class="ref" href="jordan2c.html#T22" target="_self">Th22</a>, <a class="txt" href="jordan2c.html#E37:104"><i><font color="Green">E167</font></i></a></i>;<br/>


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