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

<span class="kw">let </span><font color="Maroon" title="c1">f</font> be  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">V8</a>()  <a href="goboard5.html#V2" title="GOBOARD5:attr.2">standard</a>  <a href="goboard5.html#NM1" title="GOBOARD5:NM.1">special_circular_sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c1">f</font> <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> <font color="Maroon" title="c1">f</font></span>)</span> or <font color="Maroon" title="c1">f</font> is  <a href="sprect_2.html#V1" title="SPRECT_2:attr.1">clockwise_oriented</a>  or  <a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c1">f</font> is  <a href="sprect_2.html#V1" title="SPRECT_2:attr.1">clockwise_oriented</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:46"/><span class="lab"><font color="Green" title="E35">A1</font></span>: 
<font color="Maroon" title="c1">f</font> <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> <font color="Maroon" title="c1">f</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">f</font> is  <a href="sprect_2.html#V1" title="SPRECT_2:attr.1">clockwise_oriented</a>  or  <a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c1">f</font> is  <a href="sprect_2.html#V1" title="SPRECT_2:attr.1">clockwise_oriented</a>  )</span><br/>

<a NAME="E2:46"/>
<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="goboard7.html#T36" title="GOBOARD7:th.36">GOBOARD7:36</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E3:46"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A2</font></span>: 
2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T52" title="NAT_D:th.52">NAT_D:52</a></span>;<br/>
<a NAME="E4:46"/><span class="lab"><font color="Green" title="E37">A3</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> <font color="Maroon" title="c1">f</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
;<br/>
<a NAME="E5:46"/><span class="lab"><font color="Green" title="E38">A4</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="jordan5d.html#K7" title="JORDAN5D:func.7">i_w_n</a> <font color="Maroon" title="c1">f</font></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> <font color="Maroon" title="c1">f</font></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> <font color="Maroon" title="c1">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan5d.html#D7" title="JORDAN5D:def.7">JORDAN5D:def 7</a></span>;<br/>
<a NAME="E6:46"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A5</font></span>: 
 <a href="jordan5d.html#K7" title="JORDAN5D:func.7">i_w_n</a> <font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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> <font color="Maroon" title="c1">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/>
<a NAME="E7:46"/><span class="lab"><font color="Green" title="E40">A6</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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> <font color="Maroon" title="c1">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46"><span class="lab"><font color="Green" title="E38">A4</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/>
<a NAME="E8:46"/><span class="lab"><font color="Green" title="E41">A7</font></span>: 
<span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><a href="jordan5d.html#K7" title="JORDAN5D:func.7">i_w_n</a> <font color="Maroon" title="c1">f</font></span>)</span>,<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> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <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> <font color="Maroon" title="c1">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan5d.html#D7" title="JORDAN5D:def.7">JORDAN5D:def 7</a></span>;<br/>
<a NAME="E9:46"/><span class="lab"><font color="Green" title="E42">A8</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="goboard7.html#T36" title="GOBOARD7:th.36">GOBOARD7:36</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E10:46"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A9</font></span>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 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> <font color="Maroon" title="c1">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c2">i1</font>, <font color="Maroon" title="c3">j1</font> being    <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"> such that </span><br/><a NAME="E12:46"/><span class="lab"><font color="Green" title="E44">A10</font></span>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c3">j1</font></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> <font color="Maroon" title="c1">f</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E13:46"/><span class="lab"><font color="Green" title="E45">A11</font></span>: 
<font color="Maroon" title="c1">f</font> <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> <font color="Maroon" title="c1">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c3">j1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="goboard5.html#T12" title="GOBOARD5:th.12">GOBOARD5:12</a></span>;<br/>
<a NAME="E14:46"/><span class="lab"><font color="Green" title="E46">A12</font></span>: 
<font color="Maroon" title="c3">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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> <font color="Maroon" title="c1">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:46"><span class="lab"><font color="Green" title="E44">A10</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/>
<a NAME="E15:46"/><span class="lab"><font color="Green" title="E47">A13</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">j1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:46"><span class="lab"><font color="Green" title="E44">A10</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/>
<a NAME="E16:46"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A14</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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> <font color="Maroon" title="c1">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:46"><span class="lab"><font color="Green" title="E46">A12</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E17:46"/><span class="lab"><font color="Green" title="E49">A15</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:46"><span class="lab"><font color="Green" title="E44">A10</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/>
<a NAME="E18:46"/><span class="lab"><font color="Green" title="E50">A16</font></span>: 
<font color="Maroon" title="c2">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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> <font color="Maroon" title="c1">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:46"><span class="lab"><font color="Green" title="E44">A10</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E51">A17</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:46_1"/><span class="lab"><font color="Green" title="E52">A18</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> <font color="Maroon" title="c1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">j1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="jordan5d.html#K7" title="JORDAN5D:func.7">i_w_n</a> <font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i1</font></span><br/><a NAME="E2:46_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c3">j1</font></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> <font color="Maroon" title="c1">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan5d.html#T42" title="JORDAN5D:th.42">JORDAN5D:42</a></span>;<br/><a NAME="E3:46_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c3">j1</font></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> <font color="Maroon" title="c1">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:46"><span class="lab"><font color="Green" title="E47">A13</font></span></a>, <a class="txt" href="#E14:46"><span class="lab"><font color="Green" title="E46">A12</font></span></a>, <a class="txt" href="#E17:46"><span class="lab"><font color="Green" title="E49">A15</font></span></a>, <a class="txt" href="#E18:46"><span class="lab"><font color="Green" title="E50">A16</font></span></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></span>;<br/><a NAME="E4:46_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c3">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">N-most</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:46"><span class="lab"><font color="Green" title="E42">A8</font></span></a>, <a class="txt" href="#E10:46"><span class="lab"><font color="Green" title="E43">A9</font></span></a>, <a class="txt" href="#E13:46"><span class="lab"><font color="Green" title="E45">A11</font></span></a>, <a class="ref" href="goboard1.html#T16" title="GOBOARD1:th.16">GOBOARD1:16</a>, <a class="ref" href="sprect_2.html#T14" title="SPRECT_2:th.14">SPRECT_2:14</a></span>;<br/><a NAME="E5:46_1"/><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> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c3">j1</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="pscomp_1.html#T98" title="PSCOMP_1:th.98">PSCOMP_1:98</a></span>;<br/><span class="kw">hence </span><a NAME="E6:46_1"/>
 <a href="jordan5d.html#K7" title="JORDAN5D:func.7">i_w_n</a> <font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:46"><span class="lab"><font color="Green" title="E41">A7</font></span></a>, <a class="txt" href="#E15:46"><span class="lab"><font color="Green" title="E47">A13</font></span></a>, <a class="txt" href="#E6:46"><span class="lab"><font color="Green" title="E39">A5</font></span></a>, <a class="txt" href="#E17:46"><span class="lab"><font color="Green" title="E49">A15</font></span></a>, <a class="txt" href="#E1:46_1"><span class="lab"><font color="Green" title="E52">A18</font></span></a>, <a class="ref" href="goboard5.html#T4" title="GOBOARD5:th.4">GOBOARD5:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E20:46"/><span class="lab"><font color="Green" title="E52">A19</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="goboard7.html#T36" title="GOBOARD7:th.36">GOBOARD7:36</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E21:46"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A20</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</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> <font color="Maroon" title="c1">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E22:46"/>
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> <font color="Maroon" title="c1">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:46"><span class="lab"><font color="Green" title="E52">A19</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E23:46"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan5d.html#K7" title="JORDAN5D:func.7">i_w_n</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">i1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><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> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">j1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E5:46"><span class="lab"><font color="Green" title="E38">A4</font></span></a>, <a class="txt" href="#E8:46"><span class="lab"><font color="Green" title="E41">A7</font></span></a>, <a class="txt" href="#E10:46"><span class="lab"><font color="Green" title="E43">A9</font></span></a>, <a class="txt" href="#E12:46"><span class="lab"><font color="Green" title="E44">A10</font></span></a>, <a class="txt" href="#E13:46"><span class="lab"><font color="Green" title="E45">A11</font></span></a>, <a class="ref" href="goboard5.html#T13" title="GOBOARD5:th.13">GOBOARD5:13</a></span>;<br/>
<a NAME="E24:46"/><span class="kw">then </span>
( (  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan5d.html#K7" title="JORDAN5D:func.7">i_w_n</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">i1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &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> <font color="Maroon" title="c1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">j1</font> ) or (  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><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> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">j1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="jordan5d.html#K7" title="JORDAN5D:func.7">i_w_n</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i1</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seqm_3.html#T82" title="SEQM_3:th.82">SEQM_3:82</a></span>;<br/>
<a NAME="E25:46"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A21</font></span>: 
( ( <font color="Maroon" title="c2">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan5d.html#K7" title="JORDAN5D:func.7">i_w_n</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &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> <font color="Maroon" title="c1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">j1</font> ) or (  <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> <font color="Maroon" title="c1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp;  <a href="jordan5d.html#K7" title="JORDAN5D:func.7">i_w_n</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i1</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="txt" href="#E14:46"><span class="lab"><font color="Green" title="E46">A12</font></span></a>, <a class="txt" href="#E19:46"><span class="lab"><font color="Green" title="E51">A17</font></span></a>, <a class="ref" href="seqm_3.html#T81" title="SEQM_3:th.81">SEQM_3:81</a></span>;<br/>
<a NAME="E26:46"/>
 <a href="jordan5d.html#K8" title="JORDAN5D:func.8">i_e_n</a> <font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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> <font color="Maroon" title="c1">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan5d.html#T47" title="JORDAN5D:th.47">JORDAN5D:47</a></span>;<br/>
<a NAME="E27:46"/><span class="kw">then </span>
 <a href="jordan5d.html#K7" title="JORDAN5D:func.7">i_w_n</a> <font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <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> <font color="Maroon" title="c1">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sprect_3.html#T44" title="SPRECT_3:th.44">SPRECT_3:44</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E28:46"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A22</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan5d.html#K7" title="JORDAN5D:func.7">i_w_n</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="jordan5d.html#K7" title="JORDAN5D:func.7">i_w_n</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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> <font color="Maroon" title="c1">f</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E29:46"/><span class="lab"><font color="Green" title="E56">A23</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T44" title="NAT_D:th.44">NAT_D:44</a></span>;<br/>
<a NAME="E30:46"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:46"><span class="lab"><font color="Green" title="E42">A8</font></span></a>, <a class="ref" href="nat_d.html#T55" title="NAT_D:th.55">NAT_D:55</a></span>;<br/>
<a NAME="E31:46"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A24</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 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> <font color="Maroon" title="c1">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:46"><span class="lab"><font color="Green" title="E56">A23</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">i2</font>, <font color="Maroon" title="c5">j2</font> being    <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"> such that </span><br/><a NAME="E33:46"/><span class="lab"><font color="Green" title="E58">A25</font></span>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">i2</font>,<font color="Maroon" title="c5">j2</font></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> <font color="Maroon" title="c1">f</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E34:46"/><span class="lab"><font color="Green" title="E59">A26</font></span>: 
<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <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> <font color="Maroon" title="c1">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">i2</font>,<font color="Maroon" title="c5">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="goboard5.html#T12" title="GOBOARD5:th.12">GOBOARD5:12</a></span>;<br/>
<a NAME="E35:46"/><span class="lab"><font color="Green" title="E60">A27</font></span>: 
<font color="Maroon" title="c5">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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> <font color="Maroon" title="c1">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:46"><span class="lab"><font color="Green" title="E58">A25</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/>
<a NAME="E36:46"/><span class="lab"><font color="Green" title="E61">A28</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:46"><span class="lab"><font color="Green" title="E58">A25</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/>
<a NAME="E37:46"/><span class="lab"><font color="Green" title="E62">A29</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">j2</font> &amp; <font color="Maroon" title="c4">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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> <font color="Maroon" title="c1">f</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:46"><span class="lab"><font color="Green" title="E58">A25</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E63">A30</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="46_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:46_2"/><span class="lab"><font color="Green" title="E64">A31</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> <font color="Maroon" title="c1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">j2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="jordan5d.html#K7" title="JORDAN5D:func.7">i_w_n</a> <font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i2</font></span><br/><a NAME="E2:46_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c5">j2</font></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> <font color="Maroon" title="c1">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan5d.html#T42" title="JORDAN5D:th.42">JORDAN5D:42</a></span>;<br/><a NAME="E3:46_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">i2</font>,<font color="Maroon" title="c5">j2</font></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> <font color="Maroon" title="c1">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:46"><span class="lab"><font color="Green" title="E60">A27</font></span></a>, <a class="txt" href="#E36:46"><span class="lab"><font color="Green" title="E61">A28</font></span></a>, <a class="txt" href="#E37:46"><span class="lab"><font color="Green" title="E62">A29</font></span></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></span>;<br/><a NAME="E4:46_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">i2</font>,<font color="Maroon" title="c5">j2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">N-most</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:46"><span class="lab"><font color="Green" title="E42">A8</font></span></a>, <a class="txt" href="#E31:46"><span class="lab"><font color="Green" title="E57">A24</font></span></a>, <a class="txt" href="#E34:46"><span class="lab"><font color="Green" title="E59">A26</font></span></a>, <a class="ref" href="goboard1.html#T16" title="GOBOARD1:th.16">GOBOARD1:16</a>, <a class="ref" href="sprect_2.html#T14" title="SPRECT_2:th.14">SPRECT_2:14</a></span>;<br/><a NAME="E5:46_2"/><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> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">i2</font>,<font color="Maroon" title="c5">j2</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="pscomp_1.html#T98" title="PSCOMP_1:th.98">PSCOMP_1:98</a></span>;<br/><span class="kw">hence </span><a NAME="E6:46_2"/>
 <a href="jordan5d.html#K7" title="JORDAN5D:func.7">i_w_n</a> <font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:46"><span class="lab"><font color="Green" title="E41">A7</font></span></a>, <a class="txt" href="#E6:46"><span class="lab"><font color="Green" title="E39">A5</font></span></a>, <a class="txt" href="#E36:46"><span class="lab"><font color="Green" title="E61">A28</font></span></a>, <a class="txt" href="#E16:46"><span class="lab"><font color="Green" title="E48">A14</font></span></a>, <a class="txt" href="#E1:46_2"><span class="lab"><font color="Green" title="E64">A31</font></span></a>, <a class="ref" href="goboard5.html#T4" title="GOBOARD5:th.4">GOBOARD5:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E39:46"/><span class="lab"><font color="Green" title="E64">A32</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 4
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="goboard7.html#T36" title="GOBOARD7:th.36">GOBOARD7:36</a></span>;<br/>
<a NAME="E40:46"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A33</font></span>: 
<span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">i2</font>,<font color="Maroon" title="c5">j2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:46"><span class="lab"><font color="Green" title="E57">A24</font></span></a>, <a class="txt" href="#E34:46"><span class="lab"><font color="Green" title="E59">A26</font></span></a>, <a class="ref" href="goboard1.html#T16" title="GOBOARD1:th.16">GOBOARD1:16</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E41:46"/><span class="lab"><font color="Green" title="E66">A34</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:46"><span class="lab"><font color="Green" title="E64">A32</font></span></a>, <a class="ref" href="xreal_1.html#T237" title="XREAL_1:th.237">XREAL_1:237</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E42:46"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <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> <font color="Maroon" title="c1">f</font> <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="E43:46"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="jordan5d.html#K7" title="JORDAN5D:func.7">i_w_n</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">j2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E31:46"><span class="lab"><font color="Green" title="E57">A24</font></span></a>, <a class="txt" href="#E5:46"><span class="lab"><font color="Green" title="E38">A4</font></span></a>, <a class="txt" href="#E8:46"><span class="lab"><font color="Green" title="E41">A7</font></span></a>, <a class="txt" href="#E33:46"><span class="lab"><font color="Green" title="E58">A25</font></span></a>, <a class="txt" href="#E34:46"><span class="lab"><font color="Green" title="E59">A26</font></span></a>, <a class="txt" href="#E21:46"><span class="lab"><font color="Green" title="E53">A20</font></span></a>, <a class="txt" href="#E41:46"><span class="lab"><font color="Green" title="E66">A34</font></span></a>, <a class="ref" href="goboard5.html#T13" title="GOBOARD5:th.13">GOBOARD5:13</a></span>;<br/>
<a NAME="E44:46"/><span class="kw">then </span>
( (  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="jordan5d.html#K7" title="JORDAN5D:func.7">i_w_n</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c5">j2</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#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">f</font></span>)</span> ) or (  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">j2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <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> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c4">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan5d.html#K7" title="JORDAN5D:func.7">i_w_n</a> <font color="Maroon" title="c1">f</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seqm_3.html#T82" title="SEQM_3:th.82">SEQM_3:82</a></span>;<br/>
<a NAME="E45:46"/><span class="kw">then </span>
( ( <font color="Maroon" title="c4">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan5d.html#K7" title="JORDAN5D:func.7">i_w_n</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c5">j2</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#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">f</font></span>)</span> ) or ( <font color="Maroon" title="c5">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">f</font></span>)</span> &amp; <font color="Maroon" title="c4">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan5d.html#K7" title="JORDAN5D:func.7">i_w_n</a> <font color="Maroon" title="c1">f</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="txt" href="#E35:46"><span class="lab"><font color="Green" title="E60">A27</font></span></a>, <a class="txt" href="#E38:46"><span class="lab"><font color="Green" title="E63">A30</font></span></a>, <a class="ref" href="seqm_3.html#T81" title="SEQM_3:th.81">SEQM_3:81</a></span>;<br/>
<a NAME="E46:46"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</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> <font color="Maroon" title="c1">f</font></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> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</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> <font color="Maroon" title="c1">f</font></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> <font color="Maroon" title="c1">f</font></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="#E29:46"><span class="lab"><font color="Green" title="E56">A23</font></span></a>, <a class="txt" href="#E13:46"><span class="lab"><font color="Green" title="E45">A11</font></span></a>, <a class="txt" href="#E34:46"><span class="lab"><font color="Green" title="E59">A26</font></span></a>, <a class="txt" href="#E7:46"><span class="lab"><font color="Green" title="E40">A6</font></span></a>, <a class="txt" href="#E25:46"><span class="lab"><font color="Green" title="E54">A21</font></span></a>, <a class="txt" href="#E28:46"><span class="lab"><font color="Green" title="E55">A22</font></span></a>, <a class="txt" href="#E3:46"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a>, <a class="ref" href="goboard7.html#T39" title="GOBOARD7:th.39">GOBOARD7:39</a></span>;<br/>
<a NAME="E47:46"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</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> <font color="Maroon" title="c1">f</font></span>)</span> or <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</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> <font color="Maroon" title="c1">f</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan5d.html#T42" title="JORDAN5D:th.42">JORDAN5D:42</a></span>;<br/>
<a NAME="E48:46"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A35</font></span>: 
( <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">N-most</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> or <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">N-most</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:46"><span class="lab"><font color="Green" title="E42">A8</font></span></a>, <a class="txt" href="#E10:46"><span class="lab"><font color="Green" title="E43">A9</font></span></a>, <a class="txt" href="#E34:46"><span class="lab"><font color="Green" title="E59">A26</font></span></a>, <a class="txt" href="#E40:46"><span class="lab"><font color="Green" title="E65">A33</font></span></a>, <a class="ref" href="goboard1.html#T16" title="GOBOARD1:th.16">GOBOARD1:16</a>, <a class="ref" href="sprect_2.html#T14" title="SPRECT_2:th.14">SPRECT_2:14</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">A</font> =  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c1">f</font></span>)</span> as  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>  <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> ;<br/>
<a NAME="E50:46"/><span class="lab"><font color="Green" title="E68">A36</font></span>: 
<font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sppol_2.html#T22" title="SPPOL_2:th.22">SPPOL_2:22</a></span>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1

<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span> <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="#E39:46"><span class="lab"><font color="Green" title="E64">A32</font></span></a>, <a class="ref" href="xreal_1.html#T237" title="XREAL_1:th.237">XREAL_1:237</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>
;<br/>
<a NAME="E52:46"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A37</font></span>: 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:46"><span class="lab"><font color="Green" title="E57">A24</font></span></a>, <a class="ref" href="finseq_5.html#T69" title="FINSEQ_5:th.69">FINSEQ_5:69</a></span>;<br/>
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T68" title="FINSEQ_5:th.68">FINSEQ_5:68</a></span>
<br/>.= 
 <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> <font color="Maroon" title="c1">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="ref" href="finseq_6.html#D1" title="FINSEQ_6:def.1">FINSEQ_6:def 1</a></span>
<br/>.= 
 <a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <font color="Maroon" title="c6">A</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="sppol_2.html#T22" title="SPPOL_2:th.22">SPPOL_2:22</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E54:46"/>
( <font color="Maroon" title="c1">f</font> is  <a href="sprect_2.html#V1" title="SPRECT_2:attr.1">clockwise_oriented</a>  or  <a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c1">f</font> is  <a href="sprect_2.html#V1" title="SPRECT_2:attr.1">clockwise_oriented</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E52:46"><span class="lab"><font color="Green" title="E69">A37</font></span></a>, <a class="txt" href="#E50:46"><span class="lab"><font color="Green" title="E68">A36</font></span></a>, <a class="txt" href="#E48:46"><span class="lab"><font color="Green" title="E67">A35</font></span></a>, <a class="ref" href="sprect_2.html#T34" title="SPRECT_2:th.34">SPRECT_2:34</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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