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<b>let </b><font color="Maroon" title="c1">f</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">G</font> =  <a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c1">f</font>;<br/>
<a NAME="E1:30"/><i><font color="Green" title="E19">A1</font></i>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<b>by </b><i><a class="ref" href="goboard7.html#T35" title="GOBOARD7:th.35">GOBOARD7:35</a></i>;<br/>
<a NAME="E2:30"/><i><font color="Green" title="E20">A2</font></i>: 
 <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#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c1">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan5d.html#T47" title="JORDAN5D:th.47">JORDAN5D:47</a></i>;<br/>
<a NAME="E3:30"/><i><font color="Green" title="E21">A3</font></i>: 
<span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">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#K3" title="GOBOARD2:func.3">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#K29" title="PSCOMP_1:func.29">N-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan5d.html#D7" title="JORDAN5D:def.7">JORDAN5D:def 7</a></i>;<br/>
<a NAME="E4:30"/><i><font color="Green" title="E22">A4</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="jordan5d.html#K8" title="JORDAN5D:func.8">i_e_n</a> <font color="Maroon" title="c1">f</font>
 
<b>by </b><i><a class="ref" href="jordan5d.html#T47" title="JORDAN5D:th.47">JORDAN5D:47</a></i>;<br/>
<a NAME="E5:30"/><i><font color="Green" title="E23">A5</font></i>: 
<span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">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#K8" title="JORDAN5D:func.8">i_e_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#K3" title="GOBOARD2:func.3">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#K30" title="PSCOMP_1:func.30">N-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan5d.html#D8" title="JORDAN5D:def.8">JORDAN5D:def 8</a></i>;<br/>
<a NAME="E6:30"/><i><font color="Green" title="E24">A6</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">N-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K30" title="PSCOMP_1:func.30">N-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 
<b>by </b><i><a class="ref" href="sprect_2.html#T55" title="SPRECT_2:th.55">SPRECT_2:55</a></i>;<br/>
<a NAME="E7:30"/><b>then </b><i><font color="Green" title="E25">A7</font></i>: 
 <a href="jordan5d.html#K7" title="JORDAN5D:func.7">i_w_n</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="jordan5d.html#K8" title="JORDAN5D:func.8">i_e_n</a> <font color="Maroon" title="c1">f</font>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E5:30"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="jordan5d.html#D7" title="JORDAN5D:def.7">JORDAN5D:def 7</a></i>;<br/>
<b>assume </b><a NAME="E8:30"/>
 <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#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="jordan5d.html#K8" title="JORDAN5D:func.8">i_e_n</a> <font color="Maroon" title="c1">f</font>
 ;<br/>

<a NAME="E9:30"/><b>then </b>
 <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#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="jordan5d.html#K8" title="JORDAN5D:func.8">i_e_n</a> <font color="Maroon" title="c1">f</font>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E7:30"><i><font color="Green" title="E25">A7</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<b>hence </b><a NAME="E10:30"/>
contradiction
 <b>by </b><i><a class="txt" href="sprect_3.html#E1:30"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="sprect_3.html#E2:30"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="sprect_3.html#E3:30"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="sprect_3.html#E4:30"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="sprect_3.html#E5:30"><i><font color="Green" title="E23">A5</font></i></a>, <a class="txt" href="sprect_3.html#E6:30"><i><font color="Green" title="E24">A6</font></i></a>, <a class="ref" href="goboard5.html#T4" title="GOBOARD5:th.4">GOBOARD5:4</a></i>;<br/>


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