<?xml version="1.0"?>
<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">X</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> where <font color="Olive" title="b1">q</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">h</font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c3">A</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c1">h</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1</span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> ;<br/>
<b>consider </b><font color="Maroon" title="c4">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:26"/><i><font color="Green" title="E27">A1</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">h</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">a</font> = <font color="Maroon" title="c4">a</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="jordan5d.html#E2:26"><i><font color="Green" title="E27">A1</font></i></a></i>;<br/>
<a NAME="E4:26"/><i><font color="Green" title="E28">A2</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> where <font color="Olive" title="b1">q</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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#E2:26"><i><font color="Green" title="E27">A1</font></i></a></i>;<br/>
<a NAME="E5:26"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> where <font color="Olive" title="b1">q</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">h</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:26_1"/>
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> where <font color="Olive" title="b1">q</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">h</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">qq</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:26_1"/><i><font color="Green" title="E29">A3</font></i>: 
( <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">qq</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c7">qq</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">h</font> )
 ;<br/>
<b>thus </b><a NAME="E4:26_1"/>
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <b>by </b><i><a class="txt" href="jordan5d.html#E3:26_1"><i><font color="Green" title="E29">A3</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c6">X</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> where <font color="Olive" title="b1">q</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">h</font> </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="txt" href="jordan5d.html#E4:26"><i><font color="Green" title="E28">A2</font></i></a></i>;<br/>
<a NAME="E7:26"/>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c6">X</font> <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#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c1">h</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1</span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2"><b>proof </b></a><div class="add">

<a NAME="E1:26_2"/><i><font color="Green" title="E29">A4</font></i>: 
( 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#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c1">h</font></span>)</span> &amp; 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#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c1">h</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="goboard7.html#T34" title="GOBOARD7:th.34">GOBOARD7:34</a>, <a class="ref" href="goboard7.html#T35" title="GOBOARD7:th.35">GOBOARD7:35</a></i>;<br/>
<a NAME="E2:26_2"/><i><font color="Green" title="E30">A5</font></i>: 
for <font color="Olive" title="b1">p</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font> holds <br/><font color="Olive" title="b1">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c1">h</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1</span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">p</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><a NAME="E1:26_2_1"/>
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">s</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:26_2_1"/><i><font color="Green" title="E31">A6</font></i>: 
( <font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">h</font> )
 ;<br/>
<b>thus </b><a NAME="E4:26_2_1"/>
<font color="Maroon" title="c7">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c1">h</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1</span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan5d.html#E1:26_2"><i><font color="Green" title="E29">A4</font></i></a>, <a class="txt" href="jordan5d.html#E3:26_2_1"><i><font color="Green" title="E31">A6</font></i></a>, <a class="ref" href="jordan5d.html#T35" target="_self" title="JORDAN5D:th.35">Th35</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c7">q1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E4:26_2"/><i><font color="Green" title="E31">A7</font></i>: 
( <font color="Maroon" title="c7">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`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#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c1">h</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1</span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c7">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">h</font> )
 <b>by </b><i><a class="txt" href="jordan5d.html#E1:26_2"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="jordan5d.html#T38" target="_self" title="JORDAN5D:th.38">Th38</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">q11</font> = <font color="Maroon" title="c7">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<a NAME="E6:26_2"/>
for <font color="Olive" title="b1">q</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st ( for <font color="Olive" title="b2">p</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font> holds <br/><font color="Olive" title="b2">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">q</font> ) holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c1">h</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1</span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">q</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><a NAME="E1:26_2_2"/><i><font color="Green" title="E32">A8</font></i>: 
for <font color="Olive" title="b1">p</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font> holds <br/><font color="Olive" title="b1">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c9">q</font>
 ;<br/>

<a NAME="E2:26_2_2"/>
<font color="Maroon" title="c8">q11</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font>
 
<b>by </b><i><a class="txt" href="jordan5d.html#E4:26_2"><i><font color="Green" title="E31">A7</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:26_2_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c1">h</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1</span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c9">q</font>
 <b>by </b><i><a class="txt" href="jordan5d.html#E4:26_2"><i><font color="Green" title="E31">A7</font></i></a>, <a class="txt" href="jordan5d.html#E1:26_2_2"><i><font color="Green" title="E32">A8</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:26_2"/>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c6">X</font> <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#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c1">h</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1</span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan5d.html#E2:26_2"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="seq_4.html#T61" title="SEQ_4:th.61">SEQ_4:61</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:26"/>
 <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">S-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> <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#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c1">h</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1</span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="ref" href="jordan5d.html#T20" target="_self" title="JORDAN5D:th.20">Th20</a></i>;<br/>


</div>
