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<div class="add">

<b>let </b><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>;<br/>

<b>set </b><font color="Maroon" title="c2">i1</font> =  <a href="jordan5d.html#K9" title="JORDAN5D:func.9">n_s_w</a> <font color="Maroon" title="c1">h</font>;<br/>
<b>set </b><font color="Maroon" title="c3">i2</font> =  <a href="jordan5d.html#K11" title="JORDAN5D:func.11">n_s_e</a> <font color="Maroon" title="c1">h</font>;<br/>
<a NAME="E1:47"/><i><font color="Green" title="E59">A1</font></i>: 
(  <a href="jordan5d.html#K9" title="JORDAN5D:func.9">n_s_w</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#K9" title="JORDAN5D:func.9">n_s_w</a> <font color="Maroon" title="c1">h</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp;  <a href="jordan5d.html#K11" title="JORDAN5D:func.11">n_s_e</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#K11" title="JORDAN5D:func.11">n_s_e</a> <font color="Maroon" title="c1">h</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E2:47"/><i><font color="Green" title="E60">A2</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="jordan5d.html#K9" title="JORDAN5D:func.9">n_s_w</a> <font color="Maroon" title="c1">h</font> &amp; <span class="p1">(<span class="default"><a href="jordan5d.html#K9" title="JORDAN5D:func.9">n_s_w</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> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="jordan5d.html#K11" title="JORDAN5D:func.11">n_s_e</a> <font color="Maroon" title="c1">h</font> &amp; <span class="p1">(<span class="default"><a href="jordan5d.html#K11" title="JORDAN5D:func.11">n_s_e</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> )
 
<b>by </b><i><a class="ref" href="jordan5d.html#D9" target="_self" title="JORDAN5D:def.9">Def9</a>, <a class="ref" href="jordan5d.html#D11" target="_self" title="JORDAN5D:def.11">Def11</a></i>;<br/>
<a NAME="E3:47"/><b>then </b>
(  <a href="jordan5d.html#K9" title="JORDAN5D:func.9">n_s_w</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> &amp;  <a href="jordan5d.html#K11" title="JORDAN5D:func.11">n_s_e</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> )
 
<b>by </b><i><a class="txt" href="jordan5d.html#E1:47"><i><font color="Green" title="E59">A1</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E4:47"/><b>then </b><i><font color="Green" title="E61">A3</font></i>: 
(  <a href="jordan5d.html#K9" title="JORDAN5D:func.9">n_s_w</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> &amp;  <a href="jordan5d.html#K11" title="JORDAN5D:func.11">n_s_e</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> )
 
<b>by </b><i><a class="txt" href="jordan5d.html#E2:47"><i><font color="Green" title="E60">A2</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E5:47"/><i><font color="Green" title="E62">A4</font></i>: 
<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#K9" title="JORDAN5D:func.9">n_s_w</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#K25" title="PSCOMP_1:func.25">W-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">h</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan5d.html#D9" target="_self" title="JORDAN5D:def.9">Def9</a></i>;<br/>
<a NAME="E6:47"/><i><font color="Green" title="E63">A5</font></i>: 
<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#K11" title="JORDAN5D:func.11">n_s_e</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#K30" title="PSCOMP_1:func.30">E-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">h</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan5d.html#D11" target="_self" title="JORDAN5D:def.11">Def11</a></i>;<br/>
<a NAME="E7:47"/><i><font color="Green" title="E64">A6</font></i>: 
( <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#K9" title="JORDAN5D:func.9">n_s_w</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#K9" title="JORDAN5D:func.9">n_s_w</a> <font color="Maroon" title="c1">h</font></span>)</span> &amp; <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#K11" title="JORDAN5D:func.11">n_s_e</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#K11" title="JORDAN5D:func.11">n_s_e</a> <font color="Maroon" title="c1">h</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan5d.html#E4:47"><i><font color="Green" title="E61">A3</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<b>thus </b><a NAME="E8:47"/>
 <a href="jordan5d.html#K9" title="JORDAN5D:func.9">n_s_w</a> <font color="Maroon" title="c1">h</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="jordan5d.html#K11" title="JORDAN5D:func.11">n_s_e</a> <font color="Maroon" title="c1">h</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:47_1"/>
 <a href="jordan5d.html#K9" title="JORDAN5D:func.9">n_s_w</a> <font color="Maroon" title="c1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan5d.html#K11" title="JORDAN5D:func.11">n_s_e</a> <font color="Maroon" title="c1">h</font>
 ;<br/>

<b>then </b><i><font color="Green" title="E65">A7</font></i>:  <a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">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#K11" title="JORDAN5D:func.11">n_s_e</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
<b>by </b><i><a class="txt" href="jordan5d.html#E5:47"><i><font color="Green" title="E62">A4</font></i></a>, <a class="txt" href="jordan5d.html#E7:47"><i><font color="Green" title="E64">A6</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>
<br/>.= 
 <a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">h</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan5d.html#E6:47"><i><font color="Green" title="E63">A5</font></i></a>, <a class="txt" href="jordan5d.html#E7:47"><i><font color="Green" title="E64">A6</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>
;<br/>
<a NAME="E3:47_1"/>
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>
 
<b>by </b><i><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></i>;<br/>
<a NAME="E4:47_1"/><b>then </b>
( <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#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">h</font></span>)</span> &amp; <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#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">h</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="jordan5d.html#T14" target="_self" title="JORDAN5D:th.14">Th14</a></i>;<br/>
<a NAME="E5:47_1"/><b>then </b><i><font color="Green" title="E66">A8</font></i>: 
<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#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">h</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan5d.html#E2:47_1"><i><font color="Green" title="E65">A7</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<b>consider </b><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> <b> such that </b><br/><a NAME="E7:47_1"/><i><font color="Green" title="E67">A9</font></i>: 
( <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> &amp; <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#K21" title="EUCLID:func.21">`1</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#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="ref" href="goboard7.html#T32" title="GOBOARD7:th.32">GOBOARD7:32</a></i>;<br/>
<a NAME="E8:47_1"/><i><font color="Green" title="E68">A10</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">ii</font> &amp; <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> )
 
<b>by </b><i><a class="txt" href="jordan5d.html#E7:47_1"><i><font color="Green" title="E67">A9</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E9:47_1"/><b>then </b><i><font color="Green" title="E69">A11</font></i>: 
<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#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">h</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan5d.html#T14" target="_self" title="JORDAN5D:th.14">Th14</a></i>;<br/>
<a NAME="E10:47_1"/>
<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#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">h</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan5d.html#E8:47_1"><i><font color="Green" title="E68">A10</font></i></a>, <a class="ref" href="jordan5d.html#T14" target="_self" title="JORDAN5D:th.14">Th14</a></i>;<br/>
<b>hence </b><a NAME="E11:47_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan5d.html#E2:47_1"><i><font color="Green" title="E65">A7</font></i></a>, <a class="txt" href="jordan5d.html#E5:47_1"><i><font color="Green" title="E66">A8</font></i></a>, <a class="txt" href="jordan5d.html#E7:47_1"><i><font color="Green" title="E67">A9</font></i></a>, <a class="txt" href="jordan5d.html#E9:47_1"><i><font color="Green" title="E69">A11</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>


</div><b>end;</b></div>


</div>
