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

<span class="kw">let </span><font color="Maroon" title="c1">D</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>   non  <a href="sppol_1.html#V1" title="SPPOL_1:attr.1">horizontal</a>   non  <a href="sppol_1.html#V2" title="SPPOL_1:attr.2">vertical</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <span class="p1">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,2) &amp; <span class="p1">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (2,2) &amp; <span class="p1">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (2,1) &amp; <span class="p1">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,1) &amp; <span class="p1">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 5 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,2) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">f</font> =  <a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font>;<br/>
<a NAME="E1:93"/><span class="lab"><font color="Green" title="E81">A1</font></span>: 
 <a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sprect_1.html#T32" title="SPRECT_1:th.32">SPRECT_1:32</a></span>;<br/>
<a NAME="E2:93"/><span class="lab"><font color="Green" title="E82">A2</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sprect_1.html#T82" title="SPRECT_1:th.82">SPRECT_1:82</a></span>;<br/>
<a NAME="E3:93"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A3</font></span>: 
<span class="p1">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 5 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_6.html#D1" title="FINSEQ_6:def.1">FINSEQ_6:def 1</a></span>;<br/>
<a NAME="E4:93"/>
4 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:93"><span class="lab"><font color="Green" title="E82">A2</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E5:93"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A4</font></span>: 
4 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E6:93"/><span class="kw">then </span>
4 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sprect_2.html#T15" title="SPRECT_2:th.15">SPRECT_2:15</a></span>;<br/>
<a NAME="E7:93"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 4</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a>, <a class="ref" href="sprect_1.html#T86" title="SPRECT_1:th.86">SPRECT_1:86</a></span>;<br/>
<a NAME="E8:93"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A5</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>;<br/>
<a NAME="E9:93"/><span class="lab"><font color="Green" title="E86">A6</font></span>: 
<span class="p1">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sprect_1.html#T85" title="SPRECT_1:th.85">SPRECT_1:85</a></span>;<br/>
<a NAME="E10:93"/>
3 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:93"><span class="lab"><font color="Green" title="E82">A2</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E11:93"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A7</font></span>: 
3 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E12:93"/><span class="kw">then </span>
3 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sprect_2.html#T15" title="SPRECT_2:th.15">SPRECT_2:15</a></span>;<br/>
<a NAME="E13:93"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">E-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 3</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a>, <a class="ref" href="sprect_1.html#T85" title="SPRECT_1:th.85">SPRECT_1:85</a></span>;<br/>
<a NAME="E14:93"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A8</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>;<br/>
<a NAME="E15:93"/><span class="lab"><font color="Green" title="E89">A9</font></span>: 
<span class="p1">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K21" title="PSCOMP_1:func.21">N-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sprect_1.html#T84" title="SPRECT_1:th.84">SPRECT_1:84</a></span>;<br/>
<a NAME="E16:93"/>
3 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:93"><span class="lab"><font color="Green" title="E87">A7</font></span></a>, <a class="ref" href="sprect_2.html#T16" title="SPRECT_2:th.16">SPRECT_2:16</a></span>;<br/>
<a NAME="E17:93"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 3</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="goboard1.html#D2" title="GOBOARD1:def.2">GOBOARD1:def 2</a></span>;<br/>
<a NAME="E18:93"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A10</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:93"><span class="lab"><font color="Green" title="E86">A6</font></span></a>, <a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>;<br/>
<a NAME="E19:93"/><span class="lab"><font color="Green" title="E91">A11</font></span>: 
<span class="p1">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sprect_1.html#T83" title="SPRECT_1:th.83">SPRECT_1:83</a></span>;<br/>
<a NAME="E20:93"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:93"><span class="lab"><font color="Green" title="E82">A2</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E21:93"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A12</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E22:93"/><span class="kw">then </span>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sprect_2.html#T16" title="SPRECT_2:th.16">SPRECT_2:16</a></span>;<br/>
<a NAME="E23:93"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="goboard1.html#D2" title="GOBOARD1:def.2">GOBOARD1:def 2</a></span>;<br/>
<a NAME="E24:93"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A13</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:93"><span class="lab"><font color="Green" title="E91">A11</font></span></a>, <a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>;<br/>
<a NAME="E25:93"/><span class="lab"><font color="Green" title="E94">A14</font></span>: 
<span class="p1">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">S-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sprect_1.html#T86" title="SPRECT_1:th.86">SPRECT_1:86</a></span>;<br/>
<a NAME="E26:93"/>
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:93"><span class="lab"><font color="Green" title="E82">A2</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E27:93"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">A15</font></span>: 
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E28:93"/><span class="kw">then </span>
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sprect_2.html#T15" title="SPRECT_2:th.15">SPRECT_2:15</a></span>;<br/>
<a NAME="E29:93"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a>, <a class="ref" href="sprect_1.html#T84" title="SPRECT_1:th.84">SPRECT_1:84</a></span>;<br/>
<a NAME="E30:93"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A16</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>;<br/>
<a NAME="E31:93"/>
4 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:93"><span class="lab"><font color="Green" title="E84">A4</font></span></a>, <a class="ref" href="sprect_2.html#T16" title="SPRECT_2:th.16">SPRECT_2:16</a></span>;<br/>
<a NAME="E32:93"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 4</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="goboard1.html#D2" title="GOBOARD1:def.2">GOBOARD1:def 2</a></span>;<br/>
<a NAME="E33:93"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A17</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:93"><span class="lab"><font color="Green" title="E94">A14</font></span></a>, <a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>;<br/>
<a NAME="E34:93"/>
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:93"><span class="lab"><font color="Green" title="E95">A15</font></span></a>, <a class="ref" href="sprect_2.html#T16" title="SPRECT_2:th.16">SPRECT_2:16</a></span>;<br/>
<a NAME="E35:93"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="goboard1.html#D2" title="GOBOARD1:def.2">GOBOARD1:def 2</a></span>;<br/>
<a NAME="E36:93"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A18</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:93"><span class="lab"><font color="Green" title="E89">A9</font></span></a>, <a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>;<br/>
<a NAME="E37:93"/><span class="lab"><font color="Green" title="E99">A19</font></span>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> or <font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:93_1"/><span class="lab"><font color="Green" title="E100">A20</font></span>: 
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="93_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:93_1_1_1"/>
( <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> or <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:93_1"><span class="lab"><font color="Green" title="E100">A20</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="93_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:93_1_1_1_1"/><span class="lab"><font color="Green" title="E101">A21</font></span>: 
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:93_1_1_1_1"/>
4 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:93"><span class="lab"><font color="Green" title="E84">A4</font></span></a>, <a class="ref" href="sprect_2.html#T16" title="SPRECT_2:th.16">SPRECT_2:16</a></span>;<br/><span class="kw">hence </span><a NAME="E3:93_1_1_1_1"/>
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:93"><span class="lab"><font color="Green" title="E97">A17</font></span></a>, <a class="txt" href="#E1:93_1_1_1_1"><span class="lab"><font color="Green" title="E101">A21</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="93_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:93_1_1_1_2"/><span class="lab"><font color="Green" title="E101">A22</font></span>: 
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:93_1_1_1_2"/>
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:93"><span class="lab"><font color="Green" title="E95">A15</font></span></a>, <a class="ref" href="sprect_2.html#T16" title="SPRECT_2:th.16">SPRECT_2:16</a></span>;<br/><span class="kw">hence </span><a NAME="E3:93_1_1_1_2"/>
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:93"><span class="lab"><font color="Green" title="E98">A18</font></span></a>, <a class="txt" href="#E1:93_1_1_1_2"><span class="lab"><font color="Green" title="E101">A22</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:93_1"/>
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E38:93"/><span class="lab"><font color="Green" title="E100">A23</font></span>: 
<span class="p1">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K19" title="PSCOMP_1:func.19">W-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sprect_1.html#T83" title="SPRECT_1:th.83">SPRECT_1:83</a></span>;<br/>
<a NAME="E39:93"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:93"><span class="lab"><font color="Green" title="E92">A12</font></span></a>, <a class="ref" href="sprect_2.html#T15" title="SPRECT_2:th.15">SPRECT_2:15</a></span>;<br/>
<a NAME="E40:93"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></span>;<br/>
<a NAME="E41:93"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A24</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:93"><span class="lab"><font color="Green" title="E100">A23</font></span></a>, <a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>;<br/>
<a NAME="E42:93"/><span class="lab"><font color="Green" title="E102">A25</font></span>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> or <font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:93_2"/><span class="lab"><font color="Green" title="E103">A26</font></span>: 
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="93_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:93_2_1_1"/>
( <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> or <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:93_2"><span class="lab"><font color="Green" title="E103">A26</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="93_2_1_1_1"><span class="kw">case </span></a><a NAME="E1:93_2_1_1_1"/><span class="lab"><font color="Green" title="E104">A27</font></span>: 
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:93_2_1_1_1"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:93"><span class="lab"><font color="Green" title="E92">A12</font></span></a>, <a class="ref" href="sprect_2.html#T15" title="SPRECT_2:th.15">SPRECT_2:15</a></span>;<br/><span class="kw">hence </span><a NAME="E3:93_2_1_1_1"/>
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:93"><span class="lab"><font color="Green" title="E101">A24</font></span></a>, <a class="txt" href="#E1:93_2_1_1_1"><span class="lab"><font color="Green" title="E104">A27</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="93_2_1_1_2"><span class="kw">case </span></a><a NAME="E1:93_2_1_1_2"/><span class="lab"><font color="Green" title="E104">A28</font></span>: 
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:93_2_1_1_2"/>
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:93"><span class="lab"><font color="Green" title="E95">A15</font></span></a>, <a class="ref" href="sprect_2.html#T15" title="SPRECT_2:th.15">SPRECT_2:15</a></span>;<br/><span class="kw">hence </span><a NAME="E3:93_2_1_1_2"/>
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:93"><span class="lab"><font color="Green" title="E96">A16</font></span></a>, <a class="txt" href="#E1:93_2_1_1_2"><span class="lab"><font color="Green" title="E104">A28</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:93_2"/>
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E43:93"/><span class="lab"><font color="Green" title="E103">A29</font></span>: 
 <a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard2.html#K1" title="GOBOARD2:func.1">GoB</a> (<span class="p1">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p2">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p2">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="goboard2.html#D2" title="GOBOARD2:def.2">GOBOARD2:def 2</a></span>;<br/>
<a NAME="E44:93"/>
5 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:93"><span class="lab"><font color="Green" title="E82">A2</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E45:93"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A30</font></span>: 
5 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E46:93"/><span class="kw">then </span>
5 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sprect_2.html#T15" title="SPRECT_2:th.15">SPRECT_2:15</a></span>;<br/>
<a NAME="E47:93"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 5 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 5</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></span>;<br/>
<a NAME="E48:93"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A31</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 5 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:93"><span class="lab"><font color="Green" title="E100">A23</font></span></a>, <a class="txt" href="#E3:93"><span class="lab"><font color="Green" title="E83">A3</font></span></a>, <a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>;<br/>
<a NAME="E49:93"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> or <font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:93_3"/>
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:93_3"/><span class="lab"><font color="Green" title="E106">A32</font></span>: 
<font color="Maroon" title="c4">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:93_3"/><span class="lab"><font color="Green" title="E107">A33</font></span>: 
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">u</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">mu</font> = <font color="Maroon" title="c4">u</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:93_3"><span class="lab"><font color="Green" title="E106">A32</font></span></a></span>;<br/>
<a NAME="E6:93_3"/>
<font color="Maroon" title="c4">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:93_3"><span class="lab"><font color="Green" title="E106">A32</font></span></a>, <a class="ref" href="sprect_2.html#T15" title="SPRECT_2:th.15">SPRECT_2:15</a></span>;<br/>
<a NAME="E7:93_3"/><span class="kw">then </span>
<font color="Maroon" title="c4">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E8:93_3"/><span class="kw">then </span>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">mu</font> &amp; <font color="Maroon" title="c5">mu</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 5 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:93"><span class="lab"><font color="Green" title="E82">A2</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E9:93_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E108">A34</font></span>: 
( <font color="Maroon" title="c5">mu</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c5">mu</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c5">mu</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 or <font color="Maroon" title="c5">mu</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3 or <font color="Maroon" title="c5">mu</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan2c.html#T4" target="_self" title="JORDAN2C:th.4">Th4</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="93_3_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:93_3_1"/>
( <font color="Maroon" title="c5">mu</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c5">mu</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c5">mu</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or <font color="Maroon" title="c5">mu</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 or <font color="Maroon" title="c5">mu</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:93_3"><span class="lab"><font color="Green" title="E108">A34</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="93_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:93_3_1_1"/>
<font color="Maroon" title="c5">mu</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:93_3_1_1"/>
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:93"><span class="lab"><font color="Green" title="E101">A24</font></span></a>, <a class="txt" href="#E4:93_3"><span class="lab"><font color="Green" title="E107">A33</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="93_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:93_3_1_2"/>
<font color="Maroon" title="c5">mu</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:93_3_1_2"/>
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:93"><span class="lab"><font color="Green" title="E96">A16</font></span></a>, <a class="txt" href="#E4:93_3"><span class="lab"><font color="Green" title="E107">A33</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="93_3_1_3"><span class="kw">suppose </span></a><a NAME="E1:93_3_1_3"/>
<font color="Maroon" title="c5">mu</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:93_3_1_3"/>
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:93"><span class="lab"><font color="Green" title="E88">A8</font></span></a>, <a class="txt" href="#E4:93_3"><span class="lab"><font color="Green" title="E107">A33</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="93_3_1_4"><span class="kw">suppose </span></a><a NAME="E1:93_3_1_4"/>
<font color="Maroon" title="c5">mu</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:93_3_1_4"/>
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:93"><span class="lab"><font color="Green" title="E85">A5</font></span></a>, <a class="txt" href="#E4:93_3"><span class="lab"><font color="Green" title="E107">A33</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="93_3_1_5"><span class="kw">suppose </span></a><a NAME="E1:93_3_1_5"/>
<font color="Maroon" title="c5">mu</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:93_3_1_5"/>
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:93"><span class="lab"><font color="Green" title="E105">A31</font></span></a>, <a class="txt" href="#E4:93_3"><span class="lab"><font color="Green" title="E107">A33</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E50:93"/><span class="kw">then </span><span class="lab"><font color="Green" title="E106">A35</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:93"><span class="lab"><font color="Green" title="E102">A25</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E51:93"/><span class="kw">then </span><span class="lab"><font color="Green" title="E107">A36</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p2">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#D21" title="SEQ_4:def.21">SEQ_4:def 21</a></span>;<br/>
<a NAME="E52:93"/>
5 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:93"><span class="lab"><font color="Green" title="E104">A30</font></span></a>, <a class="ref" href="sprect_2.html#T16" title="SPRECT_2:th.16">SPRECT_2:16</a></span>;<br/>
<a NAME="E53:93"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 5 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 5</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="goboard1.html#D2" title="GOBOARD1:def.2">GOBOARD1:def 2</a></span>;<br/>
<a NAME="E54:93"/><span class="kw">then </span><span class="lab"><font color="Green" title="E108">A37</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 5 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:93"><span class="lab"><font color="Green" title="E91">A11</font></span></a>, <a class="txt" href="#E3:93"><span class="lab"><font color="Green" title="E83">A3</font></span></a>, <a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>;<br/>
<a NAME="E55:93"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> or <font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:93_4"/>
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:93_4"/><span class="lab"><font color="Green" title="E109">A38</font></span>: 
<font color="Maroon" title="c4">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:93_4"/><span class="lab"><font color="Green" title="E110">A39</font></span>: 
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">u</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">mu</font> = <font color="Maroon" title="c4">u</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:93_4"><span class="lab"><font color="Green" title="E109">A38</font></span></a></span>;<br/>
<a NAME="E6:93_4"/>
<font color="Maroon" title="c4">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:93_4"><span class="lab"><font color="Green" title="E109">A38</font></span></a>, <a class="ref" href="sprect_2.html#T16" title="SPRECT_2:th.16">SPRECT_2:16</a></span>;<br/>
<a NAME="E7:93_4"/><span class="kw">then </span>
<font color="Maroon" title="c4">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E8:93_4"/><span class="kw">then </span>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">mu</font> &amp; <font color="Maroon" title="c5">mu</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 5 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:93"><span class="lab"><font color="Green" title="E82">A2</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E9:93_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E111">A40</font></span>: 
( <font color="Maroon" title="c5">mu</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c5">mu</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c5">mu</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 or <font color="Maroon" title="c5">mu</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3 or <font color="Maroon" title="c5">mu</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan2c.html#T4" target="_self" title="JORDAN2C:th.4">Th4</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="93_4_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:93_4_1"/>
( <font color="Maroon" title="c5">mu</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c5">mu</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c5">mu</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or <font color="Maroon" title="c5">mu</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 or <font color="Maroon" title="c5">mu</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:93_4"><span class="lab"><font color="Green" title="E111">A40</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="93_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:93_4_1_1"/>
<font color="Maroon" title="c5">mu</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:93_4_1_1"/>
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:93"><span class="lab"><font color="Green" title="E93">A13</font></span></a>, <a class="txt" href="#E4:93_4"><span class="lab"><font color="Green" title="E110">A39</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="93_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:93_4_1_2"/>
<font color="Maroon" title="c5">mu</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:93_4_1_2"/>
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:93"><span class="lab"><font color="Green" title="E98">A18</font></span></a>, <a class="txt" href="#E4:93_4"><span class="lab"><font color="Green" title="E110">A39</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="93_4_1_3"><span class="kw">suppose </span></a><a NAME="E1:93_4_1_3"/>
<font color="Maroon" title="c5">mu</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:93_4_1_3"/>
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:93"><span class="lab"><font color="Green" title="E90">A10</font></span></a>, <a class="txt" href="#E4:93_4"><span class="lab"><font color="Green" title="E110">A39</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="93_4_1_4"><span class="kw">suppose </span></a><a NAME="E1:93_4_1_4"/>
<font color="Maroon" title="c5">mu</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:93_4_1_4"/>
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:93"><span class="lab"><font color="Green" title="E97">A17</font></span></a>, <a class="txt" href="#E4:93_4"><span class="lab"><font color="Green" title="E110">A39</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="93_4_1_5"><span class="kw">suppose </span></a><a NAME="E1:93_4_1_5"/>
<font color="Maroon" title="c5">mu</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:93_4_1_5"/>
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E54:93"><span class="lab"><font color="Green" title="E108">A37</font></span></a>, <a class="txt" href="#E4:93_4"><span class="lab"><font color="Green" title="E110">A39</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E56:93"/><span class="kw">then </span><span class="lab"><font color="Green" title="E109">A41</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:93"><span class="lab"><font color="Green" title="E99">A19</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E57:93"/><span class="kw">then </span>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:93"><span class="lab"><font color="Green" title="E81">A1</font></span></a>, <a class="ref" href="card_2.html#T57" title="CARD_2:th.57">CARD_2:57</a></span>;<br/>
<a NAME="E58:93"/><span class="kw">then </span><span class="lab"><font color="Green" title="E110">A42</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p2">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#D21" title="SEQ_4:def.21">SEQ_4:def 21</a></span>;<br/>
<a NAME="E59:93"/><span class="lab"><font color="Green" title="E111">A43</font></span>: 
 <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sprect_1.html#T31" title="SPRECT_1:th.31">SPRECT_1:31</a></span>;<br/>
<a NAME="E60:93"/><span class="kw">then </span><span class="lab"><font color="Green" title="E112">A44</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:93"><span class="lab"><font color="Green" title="E106">A35</font></span></a>, <a class="ref" href="card_2.html#T57" title="CARD_2:th.57">CARD_2:57</a></span>;<br/>
<a NAME="E61:93"/><span class="kw">then </span><span class="lab"><font color="Green" title="E113">A45</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p2">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#D21" title="SEQ_4:def.21">SEQ_4:def 21</a></span>;<br/>
<span class="lab"><font color="Green" title="E114">A46</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> = 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="goboard2.html#T13" title="GOBOARD2:th.13">GOBOARD2:13</a></span>
<br/>.= 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E59:93"><span class="lab"><font color="Green" title="E111">A43</font></span></a>, <a class="txt" href="#E50:93"><span class="lab"><font color="Green" title="E106">A35</font></span></a>, <a class="ref" href="card_2.html#T57" title="CARD_2:th.57">CARD_2:57</a></span>
;<br/>
<a NAME="E63:93"/><span class="kw">then </span><span class="lab"><font color="Green" title="E115">A47</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<span class="lab"><font color="Green" title="E116">A48</font></span>:  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> = 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="goboard2.html#T13" title="GOBOARD2:th.13">GOBOARD2:13</a></span>
<br/>.= 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:93"><span class="lab"><font color="Green" title="E81">A1</font></span></a>, <a class="txt" href="#E56:93"><span class="lab"><font color="Green" title="E109">A41</font></span></a>, <a class="ref" href="card_2.html#T57" title="CARD_2:th.57">CARD_2:57</a></span>
;<br/>
<a NAME="E65:93"/>
 for <font color="Olive" title="b1">p</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> holds <br/> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_7"><span class="kw">proof </span></a><div class="add">

 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K1" title="GOBOARD2:func.1">GoB</a> (<span class="p2">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p3">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p3">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>)</span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p2">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="goboard2.html#D1" title="GOBOARD2:def.1">GOBOARD2:def 1</a></span>
<br/>.= 
2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E60:93"><span class="lab"><font color="Green" title="E112">A44</font></span></a>, <a class="ref" href="seq_4.html#D21" title="SEQ_4:def.21">SEQ_4:def 21</a></span>
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">s1</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E3:93_7"/><span class="lab"><font color="Green" title="E117">A49</font></span>: 
<font color="Maroon" title="c3">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="goboard2.html#K1" title="GOBOARD2:func.1">GoB</a> (<span class="p2">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p3">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p3">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:93_7"/><span class="lab"><font color="Green" title="E118">A50</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K1" title="GOBOARD2:func.1">GoB</a> (<span class="p2">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p3">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p3">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D3" title="MATRIX_1:def.3">MATRIX_1:def 3</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c4">p</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> implies  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c5">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E6:93_7"/><span class="lab"><font color="Green" title="E119">A51</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> holds <br/> ex <font color="Olive" title="b2">s</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>( <font color="Olive" title="b2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D1" title="MATRIX_1:def.1">MATRIX_1:def 1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E7:93_7"/>
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2</span><br/>

<a NAME="E8:93_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E120">A52</font></span>: 
 ex <font color="Olive" title="b1">s2</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>( <font color="Olive" title="b1">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:93_7"><span class="lab"><font color="Green" title="E119">A51</font></span></a></span>;<br/>
<a NAME="E9:93_7"/>
<font color="Maroon" title="c3">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:93_7"><span class="lab"><font color="Green" title="E117">A49</font></span></a>, <a class="ref" href="goboard2.html#D2" title="GOBOARD2:def.2">GOBOARD2:def 2</a></span>;<br/>
<a NAME="E10:93_7"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">s</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>( <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s1</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:93_7"><span class="lab"><font color="Green" title="E119">A51</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:93_7"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E64:93"><span class="lab"><font color="Green" title="E116">A48</font></span></a>, <a class="txt" href="#E4:93_7"><span class="lab"><font color="Green" title="E118">A50</font></span></a>, <a class="txt" href="#E8:93_7"><span class="lab"><font color="Green" title="E120">A52</font></span></a>, <a class="ref" href="goboard2.html#D2" title="GOBOARD2:def.2">GOBOARD2:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E66:93"/><span class="kw">then </span><span class="lab"><font color="Green" title="E117">A53</font></span>: 
 <a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span> is    <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of 2,2, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E62:93"><span class="lab"><font color="Green" title="E114">A46</font></span></a>, <a class="ref" href="matrix_1.html#D2" title="MATRIX_1:def.2">MATRIX_1:def 2</a></span>;<br/>
<a NAME="E67:93"/><span class="lab"><font color="Green" title="E118">A54</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E64:93"><span class="lab"><font color="Green" title="E116">A48</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E68:93"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">1,1</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p5">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p4">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p5">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E63:93"><span class="lab"><font color="Green" title="E115">A47</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/>
<a NAME="E69:93"/><span class="kw">then </span><span class="lab"><font color="Green" title="E119">A55</font></span>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">1,1</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E62:93"><span class="lab"><font color="Green" title="E114">A46</font></span></a>, <a class="txt" href="#E64:93"><span class="lab"><font color="Green" title="E116">A48</font></span></a>, <a class="txt" href="#E66:93"><span class="lab"><font color="Green" title="E117">A53</font></span></a>, <a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></span>;<br/>
<a NAME="E70:93"/><span class="lab"><font color="Green" title="E120">A56</font></span>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E64:93"><span class="lab"><font color="Green" title="E116">A48</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E71:93"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">1,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p5">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p4">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p5">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E63:93"><span class="lab"><font color="Green" title="E115">A47</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/>
<a NAME="E72:93"/><span class="kw">then </span><span class="lab"><font color="Green" title="E121">A57</font></span>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">1,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E62:93"><span class="lab"><font color="Green" title="E114">A46</font></span></a>, <a class="txt" href="#E64:93"><span class="lab"><font color="Green" title="E116">A48</font></span></a>, <a class="txt" href="#E66:93"><span class="lab"><font color="Green" title="E117">A53</font></span></a>, <a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></span>;<br/>
<a NAME="E73:93"/><span class="lab"><font color="Green" title="E122">A58</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E62:93"><span class="lab"><font color="Green" title="E114">A46</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E74:93"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>,1</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p5">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p4">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p5">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E67:93"><span class="lab"><font color="Green" title="E118">A54</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/>
<a NAME="E75:93"/><span class="kw">then </span><span class="lab"><font color="Green" title="E123">A59</font></span>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>,1</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E62:93"><span class="lab"><font color="Green" title="E114">A46</font></span></a>, <a class="txt" href="#E64:93"><span class="lab"><font color="Green" title="E116">A48</font></span></a>, <a class="txt" href="#E66:93"><span class="lab"><font color="Green" title="E117">A53</font></span></a>, <a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></span>;<br/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  = 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">SE-corner</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sprect_1.html#T81" title="SPRECT_1:th.81">SPRECT_1:81</a></span>
<br/>.= 
 <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <font color="Maroon" title="c1">D</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>
<br/>.= 
 <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="sprect_1.html#T61" title="SPRECT_1:th.61">SPRECT_1:61</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p2">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E59:93"><span class="lab"><font color="Green" title="E111">A43</font></span></a>, <a class="txt" href="#E51:93"><span class="lab"><font color="Green" title="E107">A36</font></span></a>, <a class="txt" href="#E61:93"><span class="lab"><font color="Green" title="E113">A45</font></span></a>, <a class="ref" href="jordan2c.html#T6" target="_self" title="JORDAN2C:th.6">Th8</a></span>
;<br/>
<a NAME="E77:93"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p5">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p5">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>;<br/>
<a NAME="E78:93"/><span class="kw">then </span><span class="lab"><font color="Green" title="E124">A60</font></span>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:93"><span class="lab"><font color="Green" title="E103">A29</font></span></a>, <a class="txt" href="#E62:93"><span class="lab"><font color="Green" title="E114">A46</font></span></a>, <a class="txt" href="#E75:93"><span class="lab"><font color="Green" title="E123">A59</font></span></a>, <a class="ref" href="goboard2.html#D1" title="GOBOARD2:def.1">GOBOARD2:def 1</a></span>;<br/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">S-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  = 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K10" title="PSCOMP_1:func.10">SW-corner</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sprect_1.html#T80" title="SPRECT_1:th.80">SPRECT_1:80</a></span>
<br/>.= 
 <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Maroon" title="c1">D</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>
<br/>.= 
 <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="sprect_1.html#T58" title="SPRECT_1:th.58">SPRECT_1:58</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p2">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E59:93"><span class="lab"><font color="Green" title="E111">A43</font></span></a>, <a class="txt" href="#E51:93"><span class="lab"><font color="Green" title="E107">A36</font></span></a>, <a class="txt" href="#E61:93"><span class="lab"><font color="Green" title="E113">A45</font></span></a>, <a class="ref" href="jordan2c.html#T6" target="_self" title="JORDAN2C:th.6">Th8</a></span>
;<br/>
<a NAME="E80:93"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">S-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p5">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p5">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>;<br/>
<a NAME="E81:93"/><span class="kw">then </span><span class="lab"><font color="Green" title="E125">A61</font></span>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">S-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,1)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:93"><span class="lab"><font color="Green" title="E103">A29</font></span></a>, <a class="txt" href="#E69:93"><span class="lab"><font color="Green" title="E119">A55</font></span></a>, <a class="ref" href="goboard2.html#D1" title="GOBOARD2:def.1">GOBOARD2:def 1</a></span>;<br/>
<a NAME="E82:93"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p5">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p4">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p5">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E73:93"><span class="lab"><font color="Green" title="E122">A58</font></span></a>, <a class="txt" href="#E70:93"><span class="lab"><font color="Green" title="E120">A56</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/>
<a NAME="E83:93"/><span class="kw">then </span><span class="lab"><font color="Green" title="E126">A62</font></span>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E62:93"><span class="lab"><font color="Green" title="E114">A46</font></span></a>, <a class="txt" href="#E64:93"><span class="lab"><font color="Green" title="E116">A48</font></span></a>, <a class="txt" href="#E66:93"><span class="lab"><font color="Green" title="E117">A53</font></span></a>, <a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></span>;<br/>
<a NAME="E84:93"/>
 <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p2">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E59:93"><span class="lab"><font color="Green" title="E111">A43</font></span></a>, <a class="txt" href="#E51:93"><span class="lab"><font color="Green" title="E107">A36</font></span></a>, <a class="txt" href="#E61:93"><span class="lab"><font color="Green" title="E113">A45</font></span></a>, <a class="ref" href="jordan2c.html#T6" target="_self" title="JORDAN2C:th.6">Th8</a></span>;<br/>
<a NAME="E85:93"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K19" title="PSCOMP_1:func.19">W-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p2">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>;<br/>
<a NAME="E86:93"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K19" title="PSCOMP_1:func.19">W-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p5">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p5">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>;<br/>
<a NAME="E87:93"/><span class="kw">then </span><span class="lab"><font color="Green" title="E127">A63</font></span>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K19" title="PSCOMP_1:func.19">W-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:93"><span class="lab"><font color="Green" title="E103">A29</font></span></a>, <a class="txt" href="#E64:93"><span class="lab"><font color="Green" title="E116">A48</font></span></a>, <a class="txt" href="#E72:93"><span class="lab"><font color="Green" title="E121">A57</font></span></a>, <a class="ref" href="goboard2.html#D1" title="GOBOARD2:def.1">GOBOARD2:def 1</a></span>;<br/>
<a NAME="E88:93"/><span class="lab"><font color="Green" title="E128">A64</font></span>: 
( <span class="p1">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 3</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 3</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> &amp; <span class="p1">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 4</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 4</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T53" title="EUCLID:th.53">EUCLID:53</a></span>;<br/>
<a NAME="E89:93"/><span class="lab"><font color="Green" title="E129">A65</font></span>: 
( <span class="p1">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> &amp; <span class="p1">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T53" title="EUCLID:th.53">EUCLID:53</a></span>;<br/>
<a NAME="E90:93"/><span class="lab"><font color="Green" title="E130">A66</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p2">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E56:93"><span class="lab"><font color="Green" title="E109">A41</font></span></a>, <a class="ref" href="seq_4.html#D21" title="SEQ_4:def.21">SEQ_4:def 21</a></span>;<br/>
<a NAME="E91:93"/><span class="kw">then </span><span class="lab"><font color="Green" title="E131">A67</font></span>: 
 <a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p2">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:93"><span class="lab"><font color="Green" title="E81">A1</font></span></a>, <a class="txt" href="#E58:93"><span class="lab"><font color="Green" title="E110">A42</font></span></a>, <a class="ref" href="jordan2c.html#T6" target="_self" title="JORDAN2C:th.6">Th8</a></span>;<br/>
<a NAME="E92:93"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p2">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>;<br/>
<a NAME="E93:93"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p5">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p5">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 2</span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>;<br/>
<a NAME="E94:93"/><span class="kw">then </span><span class="lab"><font color="Green" title="E132">A68</font></span>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:93"><span class="lab"><font color="Green" title="E103">A29</font></span></a>, <a class="txt" href="#E64:93"><span class="lab"><font color="Green" title="E116">A48</font></span></a>, <a class="txt" href="#E72:93"><span class="lab"><font color="Green" title="E121">A57</font></span></a>, <a class="ref" href="goboard2.html#D1" title="GOBOARD2:def.1">GOBOARD2:def 1</a></span>;<br/>
<a NAME="E95:93"/><span class="lab"><font color="Green" title="E133">A69</font></span>: 
 <a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p2">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:93"><span class="lab"><font color="Green" title="E81">A1</font></span></a>, <a class="txt" href="#E90:93"><span class="lab"><font color="Green" title="E130">A66</font></span></a>, <a class="txt" href="#E58:93"><span class="lab"><font color="Green" title="E110">A42</font></span></a>, <a class="ref" href="jordan2c.html#T6" target="_self" title="JORDAN2C:th.6">Th8</a></span>;<br/>
<a NAME="E96:93"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">S-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p2">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>;<br/>
<a NAME="E97:93"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">S-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p5">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p5">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>;<br/>
<a NAME="E98:93"/><span class="kw">then </span><span class="lab"><font color="Green" title="E134">A70</font></span>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">S-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,1)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:93"><span class="lab"><font color="Green" title="E103">A29</font></span></a>, <a class="txt" href="#E69:93"><span class="lab"><font color="Green" title="E119">A55</font></span></a>, <a class="ref" href="goboard2.html#D1" title="GOBOARD2:def.1">GOBOARD2:def 1</a></span>;<br/>
<a NAME="E99:93"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K21" title="PSCOMP_1:func.21">N-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>;<br/>
<a NAME="E100:93"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K21" title="PSCOMP_1:func.21">N-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p5">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p5">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 2</span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E91:93"><span class="lab"><font color="Green" title="E131">A67</font></span></a>, <a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>;<br/>
<a NAME="E101:93"/><span class="kw">then </span><span class="lab"><font color="Green" title="E135">A71</font></span>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K21" title="PSCOMP_1:func.21">N-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:93"><span class="lab"><font color="Green" title="E103">A29</font></span></a>, <a class="txt" href="#E62:93"><span class="lab"><font color="Green" title="E114">A46</font></span></a>, <a class="txt" href="#E64:93"><span class="lab"><font color="Green" title="E116">A48</font></span></a>, <a class="txt" href="#E83:93"><span class="lab"><font color="Green" title="E126">A62</font></span></a>, <a class="ref" href="goboard2.html#D1" title="GOBOARD2:def.1">GOBOARD2:def 1</a></span>;<br/>
<a NAME="E102:93"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p2">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E95:93"><span class="lab"><font color="Green" title="E133">A69</font></span></a>, <a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>;<br/>
<a NAME="E103:93"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p5">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p5">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>;<br/>
<a NAME="E104:93"/><span class="kw">then </span><span class="lab"><font color="Green" title="E136">A72</font></span>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>,1)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:93"><span class="lab"><font color="Green" title="E103">A29</font></span></a>, <a class="txt" href="#E62:93"><span class="lab"><font color="Green" title="E114">A46</font></span></a>, <a class="txt" href="#E75:93"><span class="lab"><font color="Green" title="E123">A59</font></span></a>, <a class="ref" href="goboard2.html#D1" title="GOBOARD2:def.1">GOBOARD2:def 1</a></span>;<br/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K21" title="PSCOMP_1:func.21">N-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  = 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">NE-corner</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sprect_1.html#T77" title="SPRECT_1:th.77">SPRECT_1:77</a></span>
<br/>.= 
 <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <font color="Maroon" title="c1">D</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>
<br/>.= 
 <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="sprect_1.html#T61" title="SPRECT_1:th.61">SPRECT_1:61</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p2">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E59:93"><span class="lab"><font color="Green" title="E111">A43</font></span></a>, <a class="txt" href="#E51:93"><span class="lab"><font color="Green" title="E107">A36</font></span></a>, <a class="txt" href="#E61:93"><span class="lab"><font color="Green" title="E113">A45</font></span></a>, <a class="ref" href="jordan2c.html#T6" target="_self" title="JORDAN2C:th.6">Th8</a></span>
;<br/>
<a NAME="E106:93"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K21" title="PSCOMP_1:func.21">N-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p5">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p5">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>;<br/>
<a NAME="E107:93"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K21" title="PSCOMP_1:func.21">N-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p4">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:93"><span class="lab"><font color="Green" title="E103">A29</font></span></a>, <a class="txt" href="#E62:93"><span class="lab"><font color="Green" title="E114">A46</font></span></a>, <a class="txt" href="#E64:93"><span class="lab"><font color="Green" title="E116">A48</font></span></a>, <a class="txt" href="#E83:93"><span class="lab"><font color="Green" title="E126">A62</font></span></a>, <a class="ref" href="goboard2.html#D1" title="GOBOARD2:def.1">GOBOARD2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E108:93"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <span class="p1">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,2) &amp; <span class="p1">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (2,2) &amp; <span class="p1">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (2,1) &amp; <span class="p1">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,1) &amp; <span class="p1">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 5 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,2) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:93"><span class="lab"><font color="Green" title="E91">A11</font></span></a>, <a class="txt" href="#E38:93"><span class="lab"><font color="Green" title="E100">A23</font></span></a>, <a class="txt" href="#E15:93"><span class="lab"><font color="Green" title="E89">A9</font></span></a>, <a class="txt" href="#E9:93"><span class="lab"><font color="Green" title="E86">A6</font></span></a>, <a class="txt" href="#E25:93"><span class="lab"><font color="Green" title="E94">A14</font></span></a>, <a class="txt" href="#E3:93"><span class="lab"><font color="Green" title="E83">A3</font></span></a>, <a class="txt" href="#E62:93"><span class="lab"><font color="Green" title="E114">A46</font></span></a>, <a class="txt" href="#E64:93"><span class="lab"><font color="Green" title="E116">A48</font></span></a>, <a class="txt" href="#E87:93"><span class="lab"><font color="Green" title="E127">A63</font></span></a>, <a class="txt" href="#E78:93"><span class="lab"><font color="Green" title="E124">A60</font></span></a>, <a class="txt" href="#E81:93"><span class="lab"><font color="Green" title="E125">A61</font></span></a>, <a class="txt" href="#E94:93"><span class="lab"><font color="Green" title="E132">A68</font></span></a>, <a class="txt" href="#E101:93"><span class="lab"><font color="Green" title="E135">A71</font></span></a>, <a class="txt" href="#E104:93"><span class="lab"><font color="Green" title="E136">A72</font></span></a>, <a class="txt" href="#E98:93"><span class="lab"><font color="Green" title="E134">A70</font></span></a>, <a class="txt" href="#E89:93"><span class="lab"><font color="Green" title="E129">A65</font></span></a>, <a class="txt" href="#E88:93"><span class="lab"><font color="Green" title="E128">A64</font></span></a>, <a class="ref" href="euclid.html#T53" title="EUCLID:th.53">EUCLID:53</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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