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

<span class="kw">let </span><font color="Maroon" title="c1">h</font> be  non  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</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">  <a href="jordan5d.html#K16" title="JORDAN5D:func.16">n_e_n</a> <font color="Maroon" title="c1">h</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="jordan5d.html#K14" title="JORDAN5D:func.14">n_e_s</a> <font color="Maroon" title="c1">h</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">i1</font> =  <a href="jordan5d.html#K16" title="JORDAN5D:func.16">n_e_n</a> <font color="Maroon" title="c1">h</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">i2</font> =  <a href="jordan5d.html#K14" title="JORDAN5D:func.14">n_e_s</a> <font color="Maroon" title="c1">h</font>;<br/>
<a NAME="E1:50"/><span class="lab"><font color="Green" title="E59">A1</font></span>: 
 <a href="jordan5d.html#K16" title="JORDAN5D:func.16">n_e_n</a> <font color="Maroon" title="c1">h</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan5d.html#K16" title="JORDAN5D:func.16">n_e_n</a> <font color="Maroon" title="c1">h</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="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<a NAME="E2:50"/><span class="lab"><font color="Green" title="E60">A2</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="jordan5d.html#K16" title="JORDAN5D:func.16">n_e_n</a> <font color="Maroon" title="c1">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan5d.html#D16" target="_self" title="JORDAN5D:def.16">Def16</a></span>;<br/>
<a NAME="E3:50"/>
<span class="p1">(<span class="default"><a href="jordan5d.html#K16" title="JORDAN5D:func.16">n_e_n</a> <font color="Maroon" title="c1">h</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> <font color="Maroon" title="c1">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan5d.html#D16" target="_self" title="JORDAN5D:def.16">Def16</a></span>;<br/>
<a NAME="E4:50"/><span class="kw">then </span>
 <a href="jordan5d.html#K16" title="JORDAN5D:func.16">n_e_n</a> <font color="Maroon" title="c1">h</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> <font color="Maroon" title="c1">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E5:50"/><span class="kw">then </span>
 <a href="jordan5d.html#K16" title="JORDAN5D:func.16">n_e_n</a> <font color="Maroon" title="c1">h</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">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50"><span class="lab"><font color="Green" title="E60">A2</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E6:50"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A3</font></span>: 
<font color="Maroon" title="c1">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="jordan5d.html#K16" title="JORDAN5D:func.16">n_e_n</a> <font color="Maroon" title="c1">h</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="jordan5d.html#K16" title="JORDAN5D:func.16">n_e_n</a> <font color="Maroon" title="c1">h</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/>
<a NAME="E7:50"/><span class="lab"><font color="Green" title="E62">A4</font></span>: 
 <a href="jordan5d.html#K14" title="JORDAN5D:func.14">n_e_s</a> <font color="Maroon" title="c1">h</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan5d.html#K14" title="JORDAN5D:func.14">n_e_s</a> <font color="Maroon" title="c1">h</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="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<a NAME="E8:50"/><span class="lab"><font color="Green" title="E63">A5</font></span>: 
<font color="Maroon" title="c1">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="jordan5d.html#K14" title="JORDAN5D:func.14">n_e_s</a> <font color="Maroon" title="c1">h</font></span>)</span> <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> <font color="Maroon" title="c1">h</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan5d.html#D14" target="_self" title="JORDAN5D:def.14">Def14</a></span>;<br/>
<a NAME="E9:50"/><span class="lab"><font color="Green" title="E64">A6</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="jordan5d.html#K14" title="JORDAN5D:func.14">n_e_s</a> <font color="Maroon" title="c1">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan5d.html#D14" target="_self" title="JORDAN5D:def.14">Def14</a></span>;<br/>
<a NAME="E10:50"/>
<span class="p1">(<span class="default"><a href="jordan5d.html#K14" title="JORDAN5D:func.14">n_e_s</a> <font color="Maroon" title="c1">h</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> <font color="Maroon" title="c1">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan5d.html#D14" target="_self" title="JORDAN5D:def.14">Def14</a></span>;<br/>
<a NAME="E11:50"/><span class="kw">then </span>
 <a href="jordan5d.html#K14" title="JORDAN5D:func.14">n_e_s</a> <font color="Maroon" title="c1">h</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> <font color="Maroon" title="c1">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:50"><span class="lab"><font color="Green" title="E62">A4</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E12:50"/><span class="kw">then </span>
 <a href="jordan5d.html#K14" title="JORDAN5D:func.14">n_e_s</a> <font color="Maroon" title="c1">h</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">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:50"><span class="lab"><font color="Green" title="E64">A6</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E13:50"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A7</font></span>: 
<font color="Maroon" title="c1">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="jordan5d.html#K14" title="JORDAN5D:func.14">n_e_s</a> <font color="Maroon" title="c1">h</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="jordan5d.html#K14" title="JORDAN5D:func.14">n_e_s</a> <font color="Maroon" title="c1">h</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/>
<a NAME="E14:50"/><span class="lab"><font color="Green" title="E66">A8</font></span>: 
<font color="Maroon" title="c1">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="jordan5d.html#K16" title="JORDAN5D:func.16">n_e_n</a> <font color="Maroon" title="c1">h</font></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> <font color="Maroon" title="c1">h</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan5d.html#D16" target="_self" title="JORDAN5D:def.16">Def16</a></span>;<br/>
<span class="kw">thus </span><a NAME="E15:50"/>
 <a href="jordan5d.html#K16" title="JORDAN5D:func.16">n_e_n</a> <font color="Maroon" title="c1">h</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="jordan5d.html#K14" title="JORDAN5D:func.14">n_e_s</a> <font color="Maroon" title="c1">h</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:50_1"/>
 <a href="jordan5d.html#K16" title="JORDAN5D:func.16">n_e_n</a> <font color="Maroon" title="c1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan5d.html#K14" title="JORDAN5D:func.14">n_e_s</a> <font color="Maroon" title="c1">h</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="lab"><font color="Green" title="E67">A9</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> <font color="Maroon" title="c1">h</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="jordan5d.html#K14" title="JORDAN5D:func.14">n_e_s</a> <font color="Maroon" title="c1">h</font></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="#E14:50"><span class="lab"><font color="Green" title="E66">A8</font></span></a>, <a class="txt" href="#E6:50"><span class="lab"><font color="Green" title="E61">A3</font></span></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></span>
<br/>.= 
 <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> <font color="Maroon" title="c1">h</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:50"><span class="lab"><font color="Green" title="E63">A5</font></span></a>, <a class="txt" href="#E13:50"><span class="lab"><font color="Green" title="E65">A7</font></span></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></span>
;<br/>
<a NAME="E3:50_1"/><span class="lab"><font color="Green" title="E68">A10</font></span>: 
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">h</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="E4:50_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A11</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</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> <font color="Maroon" title="c1">h</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan5d.html#T13" target="_self" title="JORDAN5D:th.13">Th13</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">ii</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="E6:50_1"/><span class="lab"><font color="Green" title="E70">A12</font></span>: 
<font color="Maroon" title="c4">ii</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">h</font>
 <span class="kw">and </span><br/><a NAME="E7:50_1"/><span class="lab"><font color="Green" title="E71">A13</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">ii</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">h</font> <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="goboard7.html#T33" title="GOBOARD7:th.33">GOBOARD7:33</a></span>;<br/>
<a NAME="E8:50_1"/><span class="lab"><font color="Green" title="E72">A14</font></span>: 
<font color="Maroon" title="c4">ii</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> <font color="Maroon" title="c1">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:50_1"><span class="lab"><font color="Green" title="E70">A12</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E9:50_1"/><span class="lab"><font color="Green" title="E73">A15</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">ii</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:50_1"><span class="lab"><font color="Green" title="E70">A12</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E10:50_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A16</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">ii</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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> <font color="Maroon" title="c1">h</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:50_1"><span class="lab"><font color="Green" title="E72">A14</font></span></a>, <a class="ref" href="jordan5d.html#T13" target="_self" title="JORDAN5D:th.13">Th13</a></span>;<br/>
<a NAME="E11:50_1"/><span class="lab"><font color="Green" title="E75">A17</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">ii</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</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> <font color="Maroon" title="c1">h</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:50_1"><span class="lab"><font color="Green" title="E73">A15</font></span></a>, <a class="txt" href="#E8:50_1"><span class="lab"><font color="Green" title="E72">A14</font></span></a>, <a class="ref" href="jordan5d.html#T13" target="_self" title="JORDAN5D:th.13">Th13</a></span>;<br/>
<a NAME="E12:50_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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> <font color="Maroon" title="c1">h</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50_1"><span class="lab"><font color="Green" title="E68">A10</font></span></a>, <a class="ref" href="jordan5d.html#T13" target="_self" title="JORDAN5D:th.13">Th13</a></span>;<br/>
<a NAME="E13:50_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</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">h</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50_1"><span class="lab"><font color="Green" title="E67">A9</font></span></a>, <a class="txt" href="#E4:50_1"><span class="lab"><font color="Green" title="E69">A11</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:50_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50_1"><span class="lab"><font color="Green" title="E67">A9</font></span></a>, <a class="txt" href="#E7:50_1"><span class="lab"><font color="Green" title="E71">A13</font></span></a>, <a class="txt" href="#E10:50_1"><span class="lab"><font color="Green" title="E74">A16</font></span></a>, <a class="txt" href="#E11:50_1"><span class="lab"><font color="Green" title="E75">A17</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</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>


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