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<b>let </b><font color="Maroon" title="c1">p</font> be   <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>;<br/><b>let </b><font color="Maroon" title="c2">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>let </b><font color="Maroon" title="c3">p1</font>, <font color="Maroon" title="c4">p2</font> be   <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>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:41"/><i><font color="Green" title="E40">A1</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">p</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>and </b><br/><a NAME="E2:41"/><i><font color="Green" title="E41">A2</font></i>: 
( <font color="Maroon" title="c3">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  or <font color="Maroon" title="c3">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E3:41"/><i><font color="Green" title="E42">A3</font></i>: 
( not <font color="Maroon" title="c3">p1</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="c2">f</font> &amp; not <font color="Maroon" title="c4">p2</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="c2">f</font> )
 <b>and </b><br/><a NAME="E4:41"/><i><font color="Green" title="E43">A4</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>
 ;<br/>

<a NAME="E5:41"/><i><font color="Green" title="E44">A5</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">p</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E6:41"/><b>then </b><i><font color="Green" title="E45">A6</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>
 
<b>by </b><i><a class="txt" href="jordan12.html#E1:41"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E7:41"/><i><font color="Green" title="E46">A7</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c3">p1</font>
 
<b>by </b><i><a class="txt" href="jordan12.html#E1:41"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="jordan12.html#E5:41"><i><font color="Green" title="E44">A5</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E8:41"/><i><font color="Green" title="E47">A8</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="ref" href="goboard5.html#D5" title="GOBOARD5:def.5">GOBOARD5:def 5</a></i>;<br/>
<a NAME="E9:41"/>
<font color="Maroon" title="c1">p</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="c2">f</font>
 
<b>by </b><i><a class="txt" href="jordan12.html#E1:41"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="jordan12.html#E5:41"><i><font color="Green" title="E44">A5</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">LS</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E11:41"/><i><font color="Green" title="E48">A9</font></i>: 
( <font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">LS</font> &amp; <font color="Maroon" title="c5">LS</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <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="c2">f</font> ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E13:41"/><i><font color="Green" title="E49">A10</font></i>: 
( <font color="Maroon" title="c5">LS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> &amp; <font color="Maroon" title="c6">k</font> <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="c2">f</font> )
 <b>by </b><i><a class="txt" href="jordan12.html#E11:41"><i><font color="Green" title="E48">A9</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c7">G</font> =  <a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font>;<br/>
<div><i><font color="Green" title="E50">A11</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">r</font> be   <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>;<br/><b>assume </b><a NAME="E1:41_1"/><i><font color="Green" title="E51">A12</font></i>: 
<font color="Maroon" title="c8">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:41_1"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="tops_1.html#T44" title="TOPS_1:th.44">TOPS_1:44</a></i>;<br/><b>hence </b><a NAME="E3:41_1"/>
<font color="Maroon" title="c8">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E1:41_1"><i><font color="Green" title="E51">A12</font></i></a></i>;<br/><a NAME="E4:41_1"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c2">f</font>
 <b>by </b><i><a class="txt" href="jordan12.html#E8:41"><i><font color="Green" title="E47">A8</font></i></a>, <a class="txt" href="jordan12.html#E13:41"><i><font color="Green" title="E49">A10</font></i></a>, <a class="ref" href="jordan9.html#T17" title="JORDAN9:th.17">JORDAN9:17</a></i>;<br/><b>hence </b><a NAME="E5:41_1"/>
not <font color="Maroon" title="c8">r</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="c2">f</font>
 <b>by </b><i><a class="txt" href="jordan12.html#E1:41_1"><i><font color="Green" title="E51">A12</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E51">A13</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">r</font> be   <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>;<br/><b>assume </b><a NAME="E1:41_2"/><i><font color="Green" title="E52">A14</font></i>: 
<font color="Maroon" title="c8">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:41_2"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="tops_1.html#T44" title="TOPS_1:th.44">TOPS_1:44</a></i>;<br/><b>hence </b><a NAME="E3:41_2"/>
<font color="Maroon" title="c8">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E1:41_2"><i><font color="Green" title="E52">A14</font></i></a></i>;<br/><a NAME="E4:41_2"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c2">f</font>
 <b>by </b><i><a class="txt" href="jordan12.html#E8:41"><i><font color="Green" title="E47">A8</font></i></a>, <a class="txt" href="jordan12.html#E13:41"><i><font color="Green" title="E49">A10</font></i></a>, <a class="ref" href="jordan9.html#T17" title="JORDAN9:th.17">JORDAN9:17</a></i>;<br/><b>hence </b><a NAME="E5:41_2"/>
not <font color="Maroon" title="c8">r</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="c2">f</font>
 <b>by </b><i><a class="txt" href="jordan12.html#E1:41_2"><i><font color="Green" title="E52">A14</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c8">i1</font>, <font color="Maroon" title="c9">j1</font>, <font color="Maroon" title="c10">i2</font>, <font color="Maroon" title="c11">j2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E17:41"/><i><font color="Green" title="E52">A15</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</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#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span> &amp; <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">j1</font> )
 <b>and </b><br/><a NAME="E18:41"/><i><font color="Green" title="E53">A16</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<font color="Maroon" title="c11">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</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#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span> &amp; <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <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> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i2</font>,<font color="Maroon" title="c11">j2</font> )
 <b>and </b><br/><a NAME="E19:41"/><i><font color="Green" title="E54">A17</font></i>: 
( ( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> ) or ( <font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> ) or ( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> ) or ( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan12.html#E8:41"><i><font color="Green" title="E47">A8</font></i></a>, <a class="txt" href="jordan12.html#E13:41"><i><font color="Green" title="E49">A10</font></i></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></i>;<br/>
<a NAME="E20:41"/>
<font color="Maroon" title="c6">k</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> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="txt" href="jordan12.html#E13:41"><i><font color="Green" title="E49">A10</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E21:41"/><b>then </b><i><font color="Green" title="E55">A18</font></i>: 
<font color="Maroon" title="c6">k</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="c2">f</font>
 
<b>by </b><i><a class="txt" href="jordan12.html#E13:41"><i><font color="Green" title="E49">A10</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E22:41"/><b>then </b>
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<a NAME="E23:41"/><b>then </b><i><font color="Green" title="E56">A19</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="txt" href="jordan12.html#E21:41"><i><font color="Green" title="E55">A18</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<a NAME="E24:41"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan12.html#E13:41"><i><font color="Green" title="E49">A10</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E25:41"/><b>then </b><i><font color="Green" title="E57">A20</font></i>: 
<font color="Maroon" title="c6">k</font> <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="c2">f</font>
 
<b>by </b><i><a class="txt" href="jordan12.html#E13:41"><i><font color="Green" title="E49">A10</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E26:41"/><b>then </b>
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<a NAME="E27:41"/><b>then </b><i><font color="Green" title="E58">A21</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="txt" href="jordan12.html#E25:41"><i><font color="Green" title="E57">A20</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<a NAME="E28:41"/><i><font color="Green" title="E59">A22</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">p</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="jordan12.html#E1:41"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="jordan12.html#E11:41"><i><font color="Green" title="E48">A9</font></i></a>, <a class="txt" href="jordan12.html#E13:41"><i><font color="Green" title="E49">A10</font></i></a>, <a class="ref" href="sprect_2.html#T2" title="SPRECT_2:th.2">SPRECT_2:2</a>, <a class="ref" href="topreal3.html#T26" title="TOPREAL3:th.26">TOPREAL3:26</a></i>;<br/>
<a NAME="E29:41"/>
(  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font> is <a href="sppol_1.html#V2" title="SPPOL_1:attr.2">vertical</a> or  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font> is <a href="sppol_1.html#V1" title="SPPOL_1:attr.1">horizontal</a> )
 
<b>by </b><i><a class="ref" href="sppol_1.html#T41" title="SPPOL_1:th.41">SPPOL_1:41</a></i>;<br/>
<a NAME="E30:41"/><b>then </b>
(  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> is <a href="sppol_1.html#V2" title="SPPOL_1:attr.2">vertical</a> or  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> is <a href="sppol_1.html#V1" title="SPPOL_1:attr.1">horizontal</a> )
 
<b>by </b><i><a class="txt" href="jordan12.html#E13:41"><i><font color="Green" title="E49">A10</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/>
<a NAME="E31:41"/><b>then </b><i><font color="Green" title="E60">A23</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">k</font></span>)</span> <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"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 
<b>by </b><i><a class="ref" href="sppol_1.html#T36" title="SPPOL_1:th.36">SPPOL_1:36</a>, <a class="ref" href="sppol_1.html#T37" title="SPPOL_1:th.37">SPPOL_1:37</a></i>;<br/>
<a NAME="E32:41"/><i><font color="Green" title="E61">A24</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">p</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="jordan12.html#E13:41"><i><font color="Green" title="E49">A10</font></i></a>, <a class="txt" href="jordan12.html#E28:41"><i><font color="Green" title="E59">A22</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/>
<a NAME="E33:41"/><i><font color="Green" title="E62">A25</font></i>: 
( <font color="Maroon" title="c1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p1</font> &amp; <font color="Maroon" title="c1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p2</font> &amp; <font color="Maroon" title="c1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">k</font> )
 
<b>by </b><i><a class="txt" href="jordan12.html#E1:41"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="jordan12.html#E3:41"><i><font color="Green" title="E42">A3</font></i></a>, <a class="txt" href="jordan12.html#E4:41"><i><font color="Green" title="E43">A4</font></i></a>, <a class="txt" href="jordan12.html#E5:41"><i><font color="Green" title="E44">A5</font></i></a>, <a class="txt" href="jordan12.html#E6:41"><i><font color="Green" title="E45">A6</font></i></a>, <a class="txt" href="jordan12.html#E23:41"><i><font color="Green" title="E56">A19</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E34:41"/><b>then </b><i><font color="Green" title="E63">A26</font></i>: 
( <font color="Maroon" title="c3">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  implies <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> )
 
<b>by </b><i><a class="txt" href="jordan12.html#E17:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan12.html#E18:41"><i><font color="Green" title="E53">A16</font></i></a>, <a class="txt" href="jordan12.html#E31:41"><i><font color="Green" title="E60">A23</font></i></a>, <a class="txt" href="jordan12.html#E32:41"><i><font color="Green" title="E61">A24</font></i></a>, <a class="ref" href="jordan12.html#T30" target="_self" title="JORDAN12:th.30">Th30</a>, <a class="ref" href="jordan1g.html#T7" title="JORDAN1G:th.7">JORDAN1G:7</a></i>;<br/>
<a NAME="E35:41"/><i><font color="Green" title="E64">A27</font></i>: 
( <font color="Maroon" title="c3">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  implies <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> )
 
<b>by </b><i><a class="txt" href="jordan12.html#E17:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan12.html#E18:41"><i><font color="Green" title="E53">A16</font></i></a>, <a class="txt" href="jordan12.html#E31:41"><i><font color="Green" title="E60">A23</font></i></a>, <a class="txt" href="jordan12.html#E32:41"><i><font color="Green" title="E61">A24</font></i></a>, <a class="txt" href="jordan12.html#E33:41"><i><font color="Green" title="E62">A25</font></i></a>, <a class="ref" href="jordan12.html#T30" target="_self" title="JORDAN12:th.30">Th30</a>, <a class="ref" href="jordan1g.html#T6" title="JORDAN1G:th.6">JORDAN1G:6</a></i>;<br/>
<a NAME="E36:41"/><i><font color="Green" title="E65">A28</font></i>: 
<font color="Maroon" title="c10">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#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan12.html#E18:41"><i><font color="Green" title="E53">A16</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/>
<a NAME="E37:41"/><i><font color="Green" title="E66">A29</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i2</font>
 
<b>by </b><i><a class="txt" href="jordan12.html#E18:41"><i><font color="Green" title="E53">A16</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/>
<a NAME="E38:41"/><i><font color="Green" title="E67">A30</font></i>: 
<font color="Maroon" title="c8">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#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan12.html#E17:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/>
<a NAME="E39:41"/><i><font color="Green" title="E68">A31</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i1</font>
 
<b>by </b><i><a class="txt" href="jordan12.html#E17:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/>
<a NAME="E40:41"/><i><font color="Green" title="E69">A32</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j1</font>
 
<b>by </b><i><a class="txt" href="jordan12.html#E17:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/>
<a NAME="E41:41"/><i><font color="Green" title="E70">A33</font></i>: 
<font color="Maroon" title="c9">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#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan12.html#E17:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/>
<a NAME="E42:41"/><i><font color="Green" title="E71">A34</font></i>: 
<font color="Maroon" title="c11">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#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan12.html#E18:41"><i><font color="Green" title="E53">A16</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/>
<a NAME="E43:41"/><i><font color="Green" title="E72">A35</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">j2</font>
 
<b>by </b><i><a class="txt" href="jordan12.html#E18:41"><i><font color="Green" title="E53">A16</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/>
<a NAME="E44:41"/><i><font color="Green" title="E73">A36</font></i>: 
<font color="Maroon" title="c9">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">j1</font>
 
<b>by </b><i><a class="txt" href="jordan12.html#E40:41"><i><font color="Green" title="E69">A32</font></i></a>, <a class="ref" href="jordan5b.html#T1" title="JORDAN5B:th.1">JORDAN5B:1</a></i>;<br/>
<a NAME="E45:41"/><i><font color="Green" title="E74">A37</font></i>: 
<font color="Maroon" title="c8">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">i1</font>
 
<b>by </b><i><a class="txt" href="jordan12.html#E39:41"><i><font color="Green" title="E68">A31</font></i></a>, <a class="ref" href="jordan5b.html#T1" title="JORDAN5B:th.1">JORDAN5B:1</a></i>;<br/>
<a NAME="E46:41"/><i><font color="Green" title="E75">A38</font></i>: 
<font color="Maroon" title="c10">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">i2</font>
 
<b>by </b><i><a class="txt" href="jordan12.html#E37:41"><i><font color="Green" title="E66">A29</font></i></a>, <a class="ref" href="jordan5b.html#T1" title="JORDAN5B:th.1">JORDAN5B:1</a></i>;<br/>
<a NAME="E47:41"/><i><font color="Green" title="E76">A39</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan12.html#E11:41"><i><font color="Green" title="E48">A9</font></i></a>, <a class="txt" href="jordan12.html#E13:41"><i><font color="Green" title="E49">A10</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/>
<a NAME="E48:41"/><i><font color="Green" title="E77">A40</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan12.html#E4:41"><i><font color="Green" title="E43">A4</font></i></a>, <a class="txt" href="jordan12.html#E6:41"><i><font color="Green" title="E45">A6</font></i></a>, <a class="txt" href="jordan12.html#E27:41"><i><font color="Green" title="E58">A21</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<div><i><font color="Green" title="E78">A41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_3"/><i><font color="Green" title="E79">A42</font></i>: 
( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ;<br/><a NAME="E2:41_3"/><b>then </b><i><font color="Green" title="E80">A43</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c11">j2</font> <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="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="c2">f</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan12.html#E17:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E3:41_3"/>
<font color="Maroon" title="c11">j2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">j1</font>
 <b>by </b><i><a class="txt" href="jordan12.html#E1:41_3"><i><font color="Green" title="E79">A42</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:41_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan12.html#E17:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan12.html#E18:41"><i><font color="Green" title="E53">A16</font></i></a>, <a class="txt" href="jordan12.html#E38:41"><i><font color="Green" title="E67">A30</font></i></a>, <a class="txt" href="jordan12.html#E39:41"><i><font color="Green" title="E68">A31</font></i></a>, <a class="txt" href="jordan12.html#E41:41"><i><font color="Green" title="E70">A33</font></i></a>, <a class="txt" href="jordan12.html#E43:41"><i><font color="Green" title="E72">A35</font></i></a>, <a class="txt" href="jordan12.html#E1:41_3"><i><font color="Green" title="E79">A42</font></i></a>, <a class="ref" href="goboard5.html#T5" title="GOBOARD5:th.5">GOBOARD5:5</a></i>;<br/><a NAME="E5:41_3"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan12.html#E33:41"><i><font color="Green" title="E62">A25</font></i></a>, <a class="txt" href="jordan12.html#E47:41"><i><font color="Green" title="E76">A39</font></i></a>, <a class="txt" href="jordan12.html#E48:41"><i><font color="Green" title="E77">A40</font></i></a>, <a class="ref" href="topreal6.html#T38" title="TOPREAL6:th.38">TOPREAL6:38</a></i>;<br/><b>hence </b><a NAME="E6:41_3"/>
( <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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c11">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c11">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan12.html#E17:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan12.html#E18:41"><i><font color="Green" title="E53">A16</font></i></a>, <a class="txt" href="jordan12.html#E38:41"><i><font color="Green" title="E67">A30</font></i></a>, <a class="txt" href="jordan12.html#E39:41"><i><font color="Green" title="E68">A31</font></i></a>, <a class="txt" href="jordan12.html#E42:41"><i><font color="Green" title="E71">A34</font></i></a>, <a class="txt" href="jordan12.html#E43:41"><i><font color="Green" title="E72">A35</font></i></a>, <a class="txt" href="jordan12.html#E1:41_3"><i><font color="Green" title="E79">A42</font></i></a>, <a class="txt" href="jordan12.html#E2:41_3"><i><font color="Green" title="E80">A43</font></i></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E79">A44</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_4"/><i><font color="Green" title="E80">A45</font></i>: 
( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ;<br/><a NAME="E2:41_4"/><b>then </b>
<font color="Maroon" title="c9">j1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">j2</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:41_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan12.html#E17:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan12.html#E18:41"><i><font color="Green" title="E53">A16</font></i></a>, <a class="txt" href="jordan12.html#E38:41"><i><font color="Green" title="E67">A30</font></i></a>, <a class="txt" href="jordan12.html#E39:41"><i><font color="Green" title="E68">A31</font></i></a>, <a class="txt" href="jordan12.html#E40:41"><i><font color="Green" title="E69">A32</font></i></a>, <a class="txt" href="jordan12.html#E42:41"><i><font color="Green" title="E71">A34</font></i></a>, <a class="txt" href="jordan12.html#E1:41_4"><i><font color="Green" title="E80">A45</font></i></a>, <a class="ref" href="goboard5.html#T5" title="GOBOARD5:th.5">GOBOARD5:5</a></i>;<br/><a NAME="E4:41_4"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan12.html#E33:41"><i><font color="Green" title="E62">A25</font></i></a>, <a class="txt" href="jordan12.html#E47:41"><i><font color="Green" title="E76">A39</font></i></a>, <a class="txt" href="jordan12.html#E48:41"><i><font color="Green" title="E77">A40</font></i></a>, <a class="ref" href="topreal6.html#T38" title="TOPREAL6:th.38">TOPREAL6:38</a></i>;<br/><b>hence </b><a NAME="E5:41_4"/>
( <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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c9">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan12.html#E17:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan12.html#E18:41"><i><font color="Green" title="E53">A16</font></i></a>, <a class="txt" href="jordan12.html#E37:41"><i><font color="Green" title="E66">A29</font></i></a>, <a class="txt" href="jordan12.html#E38:41"><i><font color="Green" title="E67">A30</font></i></a>, <a class="txt" href="jordan12.html#E40:41"><i><font color="Green" title="E69">A32</font></i></a>, <a class="txt" href="jordan12.html#E41:41"><i><font color="Green" title="E70">A33</font></i></a>, <a class="txt" href="jordan12.html#E42:41"><i><font color="Green" title="E71">A34</font></i></a>, <a class="txt" href="jordan12.html#E43:41"><i><font color="Green" title="E72">A35</font></i></a>, <a class="txt" href="jordan12.html#E1:41_4"><i><font color="Green" title="E80">A45</font></i></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E80">A46</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_5"/><i><font color="Green" title="E81">A47</font></i>: 
( <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> &amp; <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ;<br/><a NAME="E2:41_5"/><b>then </b>
<font color="Maroon" title="c10">i2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">i1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:41_5"/><b>then </b>
<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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i2</font>,<font color="Maroon" title="c9">j1</font></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"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="jordan12.html#E37:41"><i><font color="Green" title="E66">A29</font></i></a>, <a class="txt" href="jordan12.html#E38:41"><i><font color="Green" title="E67">A30</font></i></a>, <a class="txt" href="jordan12.html#E40:41"><i><font color="Green" title="E69">A32</font></i></a>, <a class="txt" href="jordan12.html#E41:41"><i><font color="Green" title="E70">A33</font></i></a>, <a class="ref" href="goboard5.html#T4" title="GOBOARD5:th.4">GOBOARD5:4</a></i>;<br/><a NAME="E4:41_5"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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> <font color="Maroon" title="c1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c1">p</font> <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"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="txt" href="jordan12.html#E17:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan12.html#E18:41"><i><font color="Green" title="E53">A16</font></i></a>, <a class="txt" href="jordan12.html#E33:41"><i><font color="Green" title="E62">A25</font></i></a>, <a class="txt" href="jordan12.html#E47:41"><i><font color="Green" title="E76">A39</font></i></a>, <a class="txt" href="jordan12.html#E48:41"><i><font color="Green" title="E77">A40</font></i></a>, <a class="txt" href="jordan12.html#E1:41_5"><i><font color="Green" title="E81">A47</font></i></a>, <a class="ref" href="topreal6.html#T37" title="TOPREAL6:th.37">TOPREAL6:37</a></i>;<br/><b>hence </b><a NAME="E5:41_5"/>
( <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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i2</font>,1</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> <font color="Maroon" title="c1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c1">p</font> <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"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="txt" href="jordan12.html#E17:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan12.html#E18:41"><i><font color="Green" title="E53">A16</font></i></a>, <a class="txt" href="jordan12.html#E36:41"><i><font color="Green" title="E65">A28</font></i></a>, <a class="txt" href="jordan12.html#E37:41"><i><font color="Green" title="E66">A29</font></i></a>, <a class="txt" href="jordan12.html#E38:41"><i><font color="Green" title="E67">A30</font></i></a>, <a class="txt" href="jordan12.html#E39:41"><i><font color="Green" title="E68">A31</font></i></a>, <a class="txt" href="jordan12.html#E42:41"><i><font color="Green" title="E71">A34</font></i></a>, <a class="txt" href="jordan12.html#E43:41"><i><font color="Green" title="E72">A35</font></i></a>, <a class="txt" href="jordan12.html#E1:41_5"><i><font color="Green" title="E81">A47</font></i></a>, <a class="ref" href="goboard5.html#T3" title="GOBOARD5:th.3">GOBOARD5:3</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E81">A48</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_6"/><i><font color="Green" title="E82">A49</font></i>: 
( <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> &amp; <font color="Maroon" title="c10">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ;<br/><a NAME="E2:41_6"/><b>then </b>
<font color="Maroon" title="c8">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">i2</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:41_6"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">k</font></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"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="jordan12.html#E17:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan12.html#E18:41"><i><font color="Green" title="E53">A16</font></i></a>, <a class="txt" href="jordan12.html#E36:41"><i><font color="Green" title="E65">A28</font></i></a>, <a class="txt" href="jordan12.html#E39:41"><i><font color="Green" title="E68">A31</font></i></a>, <a class="txt" href="jordan12.html#E42:41"><i><font color="Green" title="E71">A34</font></i></a>, <a class="txt" href="jordan12.html#E43:41"><i><font color="Green" title="E72">A35</font></i></a>, <a class="txt" href="jordan12.html#E1:41_6"><i><font color="Green" title="E82">A49</font></i></a>, <a class="ref" href="goboard5.html#T4" title="GOBOARD5:th.4">GOBOARD5:4</a></i>;<br/><a NAME="E4:41_6"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">k</font></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> <font color="Maroon" title="c1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c1">p</font> <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"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="txt" href="jordan12.html#E33:41"><i><font color="Green" title="E62">A25</font></i></a>, <a class="txt" href="jordan12.html#E47:41"><i><font color="Green" title="E76">A39</font></i></a>, <a class="txt" href="jordan12.html#E48:41"><i><font color="Green" title="E77">A40</font></i></a>, <a class="ref" href="topreal6.html#T37" title="TOPREAL6:th.37">TOPREAL6:37</a></i>;<br/><b>hence </b><a NAME="E5:41_6"/>
( <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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">i1</font>,1</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> <font color="Maroon" title="c1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c1">p</font> <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"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="txt" href="jordan12.html#E17:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan12.html#E18:41"><i><font color="Green" title="E53">A16</font></i></a>, <a class="txt" href="jordan12.html#E36:41"><i><font color="Green" title="E65">A28</font></i></a>, <a class="txt" href="jordan12.html#E37:41"><i><font color="Green" title="E66">A29</font></i></a>, <a class="txt" href="jordan12.html#E38:41"><i><font color="Green" title="E67">A30</font></i></a>, <a class="txt" href="jordan12.html#E39:41"><i><font color="Green" title="E68">A31</font></i></a>, <a class="txt" href="jordan12.html#E41:41"><i><font color="Green" title="E70">A33</font></i></a>, <a class="txt" href="jordan12.html#E43:41"><i><font color="Green" title="E72">A35</font></i></a>, <a class="txt" href="jordan12.html#E1:41_6"><i><font color="Green" title="E82">A49</font></i></a>, <a class="ref" href="goboard5.html#T3" title="GOBOARD5:th.3">GOBOARD5:3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E53:41"/><i><font color="Green" title="E82">A50</font></i>: 
( <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> &amp; <font color="Maroon" title="c10">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies (  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> &amp;  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c8">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> ) )
 
<b>by </b><i><a class="txt" href="jordan12.html#E8:41"><i><font color="Green" title="E47">A8</font></i></a>, <a class="txt" href="jordan12.html#E13:41"><i><font color="Green" title="E49">A10</font></i></a>, <a class="txt" href="jordan12.html#E17:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan12.html#E18:41"><i><font color="Green" title="E53">A16</font></i></a>, <a class="ref" href="gobrd13.html#T24" title="GOBRD13:th.24">GOBRD13:24</a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a></i>;<br/>
<a NAME="E54:41"/><i><font color="Green" title="E83">A51</font></i>: 
( <font color="Maroon" title="c10">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <font color="Maroon" title="c8">i1</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#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan12.html#E36:41"><i><font color="Green" title="E65">A28</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<div><i><font color="Green" title="E84">A52</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_7"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_7"/>
( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">j1</font> &amp; <font color="Maroon" title="c10">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ;<br/><a NAME="E2:41_7"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i1</font> &amp; <font color="Maroon" title="c8">i1</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#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 &amp; <font color="Maroon" title="c9">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="c2">f</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan12.html#E17:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan12.html#E36:41"><i><font color="Green" title="E65">A28</font></i></a>, <a class="txt" href="jordan12.html#E41:41"><i><font color="Green" title="E70">A33</font></i></a>, <a class="txt" href="jordan12.html#E44:41"><i><font color="Green" title="E73">A36</font></i></a>, <a class="ref" href="binarith.html#T67" title="BINARITH:th.67">BINARITH:67</a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E3:41_7"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c8">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">i1</font>,1</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> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c9">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">s</font> &amp; <font color="Olive" title="b2">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span> 
 <b>by </b><i><a class="ref" href="goboard6.html#T29" title="GOBOARD6:th.29">GOBOARD6:29</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E85">A53</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_8"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_8"/><i><font color="Green" title="E86">A54</font></i>: 
( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j1</font> &amp; <font color="Maroon" title="c10">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ;<br/><a NAME="E2:41_8"/><b>then </b>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c8">i1</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c8">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="nat_2.html#T10" title="NAT_2:th.10">NAT_2:10</a></i>;<br/><b>hence </b><a NAME="E3:41_8"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c8">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">i1</font>,1</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> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b2">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</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>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jordan12.html#E39:41"><i><font color="Green" title="E68">A31</font></i></a>, <a class="txt" href="jordan12.html#E54:41"><i><font color="Green" title="E83">A51</font></i></a>, <a class="txt" href="jordan12.html#E1:41_8"><i><font color="Green" title="E86">A54</font></i></a>, <a class="ref" href="goboard6.html#T27" title="GOBOARD6:th.27">GOBOARD6:27</a></i>;<br/></div><b>end;</b></div>
<a NAME="E57:41"/><i><font color="Green" title="E86">A55</font></i>: 
( <font color="Maroon" title="c9">j1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="c2">f</font></span>)</span> &amp; <font color="Maroon" title="c10">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">i1</font>,1</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> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">s</font> &amp; <font color="Olive" title="b2">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c9">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span>  )
 
<b>by </b><i><a class="txt" href="jordan12.html#E39:41"><i><font color="Green" title="E68">A31</font></i></a>, <a class="txt" href="jordan12.html#E40:41"><i><font color="Green" title="E69">A32</font></i></a>, <a class="txt" href="jordan12.html#E54:41"><i><font color="Green" title="E83">A51</font></i></a>, <a class="ref" href="goboard6.html#T29" title="GOBOARD6:th.29">GOBOARD6:29</a></i>;<br/>
<a NAME="E58:41"/><i><font color="Green" title="E87">A56</font></i>: 
( <font color="Maroon" title="c9">j1</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#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span> &amp; <font color="Maroon" title="c10">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">i1</font>,1</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> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <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="c2">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#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">s</font> ) </span>}</span>  )
 
<b>by </b><i><a class="txt" href="jordan12.html#E39:41"><i><font color="Green" title="E68">A31</font></i></a>, <a class="txt" href="jordan12.html#E54:41"><i><font color="Green" title="E83">A51</font></i></a>, <a class="ref" href="goboard6.html#T28" title="GOBOARD6:th.28">GOBOARD6:28</a></i>;<br/>
<a NAME="E59:41"/><i><font color="Green" title="E88">A57</font></i>: 
( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <font color="Maroon" title="c10">i2</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#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan12.html#E38:41"><i><font color="Green" title="E67">A30</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E60:41"/><i><font color="Green" title="E89">A58</font></i>: 
( <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> &amp; <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies (  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> &amp;  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">i2</font>,<font color="Maroon" title="c9">j1</font></span>)</span> ) )
 
<b>by </b><i><a class="txt" href="jordan12.html#E8:41"><i><font color="Green" title="E47">A8</font></i></a>, <a class="txt" href="jordan12.html#E13:41"><i><font color="Green" title="E49">A10</font></i></a>, <a class="txt" href="jordan12.html#E17:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan12.html#E18:41"><i><font color="Green" title="E53">A16</font></i></a>, <a class="ref" href="gobrd13.html#T26" title="GOBRD13:th.26">GOBRD13:26</a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a></i>;<br/>
<div><i><font color="Green" title="E90">A59</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_9"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_9"/>
( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">j1</font> &amp; <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ;<br/><a NAME="E2:41_9"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c10">i2</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#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 &amp; <font color="Maroon" title="c9">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="c2">f</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan12.html#E18:41"><i><font color="Green" title="E53">A16</font></i></a>, <a class="txt" href="jordan12.html#E38:41"><i><font color="Green" title="E67">A30</font></i></a>, <a class="txt" href="jordan12.html#E41:41"><i><font color="Green" title="E70">A33</font></i></a>, <a class="txt" href="jordan12.html#E44:41"><i><font color="Green" title="E73">A36</font></i></a>, <a class="ref" href="binarith.html#T67" title="BINARITH:th.67">BINARITH:67</a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E3:41_9"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i2</font>,1</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> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c9">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">s</font> &amp; <font color="Olive" title="b2">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span> 
 <b>by </b><i><a class="ref" href="goboard6.html#T29" title="GOBOARD6:th.29">GOBOARD6:29</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E91">A60</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_10"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_10"/><i><font color="Green" title="E92">A61</font></i>: 
( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j1</font> &amp; <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ;<br/><a NAME="E2:41_10"/><b>then </b>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">i2</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="nat_2.html#T10" title="NAT_2:th.10">NAT_2:10</a></i>;<br/><b>hence </b><a NAME="E3:41_10"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i2</font>,1</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> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b2">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</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>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jordan12.html#E37:41"><i><font color="Green" title="E66">A29</font></i></a>, <a class="txt" href="jordan12.html#E59:41"><i><font color="Green" title="E88">A57</font></i></a>, <a class="txt" href="jordan12.html#E1:41_10"><i><font color="Green" title="E92">A61</font></i></a>, <a class="ref" href="goboard6.html#T27" title="GOBOARD6:th.27">GOBOARD6:27</a></i>;<br/></div><b>end;</b></div>
<a NAME="E63:41"/><i><font color="Green" title="E92">A62</font></i>: 
( <font color="Maroon" title="c9">j1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="c2">f</font></span>)</span> &amp; <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">i2</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i2</font>,1</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> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">s</font> &amp; <font color="Olive" title="b2">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c9">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span>  )
 
<b>by </b><i><a class="txt" href="jordan12.html#E37:41"><i><font color="Green" title="E66">A29</font></i></a>, <a class="txt" href="jordan12.html#E40:41"><i><font color="Green" title="E69">A32</font></i></a>, <a class="txt" href="jordan12.html#E59:41"><i><font color="Green" title="E88">A57</font></i></a>, <a class="ref" href="goboard6.html#T29" title="GOBOARD6:th.29">GOBOARD6:29</a></i>;<br/>
<a NAME="E64:41"/><i><font color="Green" title="E93">A63</font></i>: 
( <font color="Maroon" title="c9">j1</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#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span> &amp; <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">i2</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i2</font>,1</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> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <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="c2">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#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">s</font> ) </span>}</span>  )
 
<b>by </b><i><a class="txt" href="jordan12.html#E37:41"><i><font color="Green" title="E66">A29</font></i></a>, <a class="txt" href="jordan12.html#E59:41"><i><font color="Green" title="E88">A57</font></i></a>, <a class="ref" href="goboard6.html#T28" title="GOBOARD6:th.28">GOBOARD6:28</a></i>;<br/>
<a NAME="E65:41"/><i><font color="Green" title="E94">A64</font></i>: 
( <font color="Maroon" title="c11">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <font color="Maroon" title="c9">j1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="c2">f</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan12.html#E42:41"><i><font color="Green" title="E71">A34</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E66:41"/><i><font color="Green" title="E95">A65</font></i>: 
( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies (  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c9">j1</font></span>)</span> &amp;  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> ) )
 
<b>by </b><i><a class="txt" href="jordan12.html#E8:41"><i><font color="Green" title="E47">A8</font></i></a>, <a class="txt" href="jordan12.html#E13:41"><i><font color="Green" title="E49">A10</font></i></a>, <a class="txt" href="jordan12.html#E17:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan12.html#E18:41"><i><font color="Green" title="E53">A16</font></i></a>, <a class="ref" href="gobrd13.html#T22" title="GOBRD13:th.22">GOBRD13:22</a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a></i>;<br/>
<div><i><font color="Green" title="E96">A66</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_11"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_11"/>
( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">i1</font> &amp; <font color="Maroon" title="c11">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ;<br/><a NAME="E2:41_11"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 &amp; <font color="Maroon" title="c8">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j1</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="c2">f</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan12.html#E17:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan12.html#E38:41"><i><font color="Green" title="E67">A30</font></i></a>, <a class="txt" href="jordan12.html#E42:41"><i><font color="Green" title="E71">A34</font></i></a>, <a class="txt" href="jordan12.html#E45:41"><i><font color="Green" title="E74">A37</font></i></a>, <a class="ref" href="binarith.html#T67" title="BINARITH:th.67">BINARITH:67</a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E3:41_11"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,1</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> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">s</font> &amp; <font color="Olive" title="b2">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c9">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span> 
 <b>by </b><i><a class="ref" href="goboard6.html#T29" title="GOBOARD6:th.29">GOBOARD6:29</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E97">A67</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_12"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_12"/><i><font color="Green" title="E98">A68</font></i>: 
( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i1</font> &amp; <font color="Maroon" title="c11">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ;<br/><a NAME="E2:41_12"/><b>then </b>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c9">j1</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_2.html#T10" title="NAT_2:th.10">NAT_2:10</a></i>;<br/><b>hence </b><a NAME="E3:41_12"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">s</font> &amp; <font color="Olive" title="b2">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c9">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jordan12.html#E40:41"><i><font color="Green" title="E69">A32</font></i></a>, <a class="txt" href="jordan12.html#E65:41"><i><font color="Green" title="E94">A64</font></i></a>, <a class="txt" href="jordan12.html#E1:41_12"><i><font color="Green" title="E98">A68</font></i></a>, <a class="ref" href="goboard6.html#T23" title="GOBOARD6:th.23">GOBOARD6:23</a></i>;<br/></div><b>end;</b></div>
<a NAME="E69:41"/><i><font color="Green" title="E98">A69</font></i>: 
( <font color="Maroon" title="c8">i1</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#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span> &amp; <font color="Maroon" title="c11">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">i1</font>,1</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> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">s</font> &amp; <font color="Olive" title="b2">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c9">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span>  )
 
<b>by </b><i><a class="txt" href="jordan12.html#E39:41"><i><font color="Green" title="E68">A31</font></i></a>, <a class="txt" href="jordan12.html#E40:41"><i><font color="Green" title="E69">A32</font></i></a>, <a class="txt" href="jordan12.html#E65:41"><i><font color="Green" title="E94">A64</font></i></a>, <a class="ref" href="goboard6.html#T29" title="GOBOARD6:th.29">GOBOARD6:29</a></i>;<br/>
<a NAME="E70:41"/><i><font color="Green" title="E99">A70</font></i>: 
( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c2">f</font></span>)</span> &amp; <font color="Maroon" title="c11">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>,1</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> <font color="Olive" title="b1">r</font> &amp; <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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">s</font> &amp; <font color="Olive" title="b2">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c9">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span>  )
 
<b>by </b><i><a class="txt" href="jordan12.html#E40:41"><i><font color="Green" title="E69">A32</font></i></a>, <a class="txt" href="jordan12.html#E65:41"><i><font color="Green" title="E94">A64</font></i></a>, <a class="ref" href="goboard6.html#T26" title="GOBOARD6:th.26">GOBOARD6:26</a></i>;<br/>
<a NAME="E71:41"/><i><font color="Green" title="E100">A71</font></i>: 
( <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <font color="Maroon" title="c11">j2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="c2">f</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan12.html#E41:41"><i><font color="Green" title="E70">A33</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E72:41"/><i><font color="Green" title="E101">A72</font></i>: 
( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies (  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c11">j2</font></span>)</span> &amp;  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">i2</font>,<font color="Maroon" title="c11">j2</font></span>)</span> ) )
 
<b>by </b><i><a class="txt" href="jordan12.html#E8:41"><i><font color="Green" title="E47">A8</font></i></a>, <a class="txt" href="jordan12.html#E13:41"><i><font color="Green" title="E49">A10</font></i></a>, <a class="txt" href="jordan12.html#E17:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan12.html#E18:41"><i><font color="Green" title="E53">A16</font></i></a>, <a class="ref" href="gobrd13.html#T28" title="GOBRD13:th.28">GOBRD13:28</a>, <a class="ref" href="gobrd13.html#T29" title="GOBRD13:th.29">GOBRD13:29</a></i>;<br/>
<div><i><font color="Green" title="E102">A73</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_13"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_13"/>
( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ;<br/><a NAME="E2:41_13"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 &amp; <font color="Maroon" title="c10">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">j2</font> &amp; <font color="Maroon" title="c11">j2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="c2">f</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan12.html#E18:41"><i><font color="Green" title="E53">A16</font></i></a>, <a class="txt" href="jordan12.html#E36:41"><i><font color="Green" title="E65">A28</font></i></a>, <a class="txt" href="jordan12.html#E41:41"><i><font color="Green" title="E70">A33</font></i></a>, <a class="txt" href="jordan12.html#E46:41"><i><font color="Green" title="E75">A38</font></i></a>, <a class="ref" href="binarith.html#T67" title="BINARITH:th.67">BINARITH:67</a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E3:41_13"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c11">j2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,1</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> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c11">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">s</font> &amp; <font color="Olive" title="b2">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c11">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span> 
 <b>by </b><i><a class="ref" href="goboard6.html#T29" title="GOBOARD6:th.29">GOBOARD6:29</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E103">A74</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_14"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_14"/><i><font color="Green" title="E104">A75</font></i>: 
( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ;<br/><a NAME="E2:41_14"/><b>then </b>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c11">j2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c11">j2</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_2.html#T10" title="NAT_2:th.10">NAT_2:10</a></i>;<br/><b>hence </b><a NAME="E3:41_14"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c11">j2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c11">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">s</font> &amp; <font color="Olive" title="b2">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c11">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jordan12.html#E43:41"><i><font color="Green" title="E72">A35</font></i></a>, <a class="txt" href="jordan12.html#E71:41"><i><font color="Green" title="E100">A71</font></i></a>, <a class="txt" href="jordan12.html#E1:41_14"><i><font color="Green" title="E104">A75</font></i></a>, <a class="ref" href="goboard6.html#T23" title="GOBOARD6:th.23">GOBOARD6:23</a></i>;<br/></div><b>end;</b></div>
<a NAME="E75:41"/><i><font color="Green" title="E104">A76</font></i>: 
( <font color="Maroon" title="c10">i2</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#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span> &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">i2</font>,<font color="Maroon" title="c11">j2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i2</font>,1</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> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c11">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">s</font> &amp; <font color="Olive" title="b2">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c11">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span>  )
 
<b>by </b><i><a class="txt" href="jordan12.html#E37:41"><i><font color="Green" title="E66">A29</font></i></a>, <a class="txt" href="jordan12.html#E43:41"><i><font color="Green" title="E72">A35</font></i></a>, <a class="txt" href="jordan12.html#E71:41"><i><font color="Green" title="E100">A71</font></i></a>, <a class="ref" href="goboard6.html#T29" title="GOBOARD6:th.29">GOBOARD6:29</a></i>;<br/>
<a NAME="E76:41"/><i><font color="Green" title="E105">A77</font></i>: 
( <font color="Maroon" title="c10">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c2">f</font></span>)</span> &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">i2</font>,<font color="Maroon" title="c11">j2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>,1</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> <font color="Olive" title="b1">r</font> &amp; <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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c11">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">s</font> &amp; <font color="Olive" title="b2">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c11">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span>  )
 
<b>by </b><i><a class="txt" href="jordan12.html#E43:41"><i><font color="Green" title="E72">A35</font></i></a>, <a class="txt" href="jordan12.html#E71:41"><i><font color="Green" title="E100">A71</font></i></a>, <a class="ref" href="goboard6.html#T26" title="GOBOARD6:th.26">GOBOARD6:26</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_15"/>
( <font color="Maroon" title="c3">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  or <font color="Maroon" title="c3">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 </span><b>by </b><i><a class="txt" href="jordan12.html#E2:41"><i><font color="Green" title="E41">A2</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_1"><b>suppose </b></a><a NAME="E1:41_15_1"/><i><font color="Green" title="E106">A78</font></i>: 
<font color="Maroon" title="c3">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 ;<br/><div class="add"><a NAME="E2:41_15_1"/><b>then </b><i><font color="Green" title="E107">A79</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan12.html#E7:41"><i><font color="Green" title="E46">A7</font></i></a>, <a class="ref" href="goboard7.html#T6" title="GOBOARD7:th.6">GOBOARD7:6</a></i>;<br/><a NAME="E3:41_15_1"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="jordan12.html#E8:41"><i><font color="Green" title="E47">A8</font></i></a>, <a class="txt" href="jordan12.html#E13:41"><i><font color="Green" title="E49">A10</font></i></a>, <a class="ref" href="jordan9.html#T11" title="JORDAN9:th.11">JORDAN9:11</a></i>;<br/><b>then consider </b><font color="Maroon" title="c12">rl'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:41_15_1"/><i><font color="Green" title="E108">A80</font></i>: 
<font color="Maroon" title="c12">rl'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 <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="c13">rl'</font> = <font color="Maroon" title="c12">rl'</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="jordan12.html#E5:41_15_1"><i><font color="Green" title="E108">A80</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c14">rl</font> = <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">rl'</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> 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> ;<br/><a NAME="E8:41_15_1"/><i><font color="Green" title="E109">A81</font></i>: 
( <font color="Maroon" title="c14">rl</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c14">rl</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">rl'</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/><a NAME="E9:41_15_1"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="jordan12.html#E8:41"><i><font color="Green" title="E47">A8</font></i></a>, <a class="txt" href="jordan12.html#E13:41"><i><font color="Green" title="E49">A10</font></i></a>, <a class="ref" href="jordan9.html#T11" title="JORDAN9:th.11">JORDAN9:11</a></i>;<br/><b>then consider </b><font color="Maroon" title="c15">rp'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E11:41_15_1"/><i><font color="Green" title="E110">A82</font></i>: 
<font color="Maroon" title="c15">rp'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 <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="c16">rp'</font> = <font color="Maroon" title="c15">rp'</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="jordan12.html#E11:41_15_1"><i><font color="Green" title="E110">A82</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c17">rp</font> = <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">rp'</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> 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> ;<br/><a NAME="E14:41_15_1"/><i><font color="Green" title="E111">A83</font></i>: 
( <font color="Maroon" title="c17">rp</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c17">rp</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">rp'</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/><div><i><font color="Green" title="E112">A84</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_15_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_15_1_1"/><i><font color="Green" title="E113">A85</font></i>: 
( <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c10">i2</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#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span> )
 ;<br/><a NAME="E2:41_15_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c13">rl'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i2</font>,1</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> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c11">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">s</font> &amp; <font color="Olive" title="b2">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c11">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan12.html#E17:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan12.html#E18:41"><i><font color="Green" title="E53">A16</font></i></a>, <a class="txt" href="jordan12.html#E31:41"><i><font color="Green" title="E60">A23</font></i></a>, <a class="txt" href="jordan12.html#E32:41"><i><font color="Green" title="E61">A24</font></i></a>, <a class="txt" href="jordan12.html#E33:41"><i><font color="Green" title="E62">A25</font></i></a>, <a class="txt" href="jordan12.html#E72:41"><i><font color="Green" title="E101">A72</font></i></a>, <a class="txt" href="jordan12.html#E75:41"><i><font color="Green" title="E104">A76</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_1"><i><font color="Green" title="E106">A78</font></i></a>, <a class="txt" href="jordan12.html#E5:41_15_1"><i><font color="Green" title="E108">A80</font></i></a>, <a class="ref" href="jordan12.html#T30" target="_self" title="JORDAN12:th.30">Th30</a>, <a class="ref" href="jordan1g.html#T7" title="JORDAN1G:th.7">JORDAN1G:7</a></i>;<br/><a NAME="E3:41_15_1_1"/><b>then </b>
( <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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i2</font>,1</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> <font color="Maroon" title="c14">rl</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c14">rl</font> <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"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c11">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c11">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan12.html#E17:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan12.html#E18:41"><i><font color="Green" title="E53">A16</font></i></a>, <a class="txt" href="jordan12.html#E31:41"><i><font color="Green" title="E60">A23</font></i></a>, <a class="txt" href="jordan12.html#E32:41"><i><font color="Green" title="E61">A24</font></i></a>, <a class="txt" href="jordan12.html#E33:41"><i><font color="Green" title="E62">A25</font></i></a>, <a class="txt" href="jordan12.html#E49:41"><i><font color="Green" title="E78">A41</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_1"><i><font color="Green" title="E106">A78</font></i></a>, <a class="txt" href="jordan12.html#E8:41_15_1"><i><font color="Green" title="E109">A81</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_1_1"><i><font color="Green" title="E113">A85</font></i></a>, <a class="ref" href="jordan12.html#T30" target="_self" title="JORDAN12:th.30">Th30</a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a>, <a class="ref" href="jordan1g.html#T7" title="JORDAN1G:th.7">JORDAN1G:7</a></i>;<br/><b>hence </b><a NAME="E4:41_15_1_1"/>
<font color="Maroon" title="c14">rl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E17:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan12.html#E18:41"><i><font color="Green" title="E53">A16</font></i></a>, <a class="txt" href="jordan12.html#E31:41"><i><font color="Green" title="E60">A23</font></i></a>, <a class="txt" href="jordan12.html#E32:41"><i><font color="Green" title="E61">A24</font></i></a>, <a class="txt" href="jordan12.html#E33:41"><i><font color="Green" title="E62">A25</font></i></a>, <a class="txt" href="jordan12.html#E72:41"><i><font color="Green" title="E101">A72</font></i></a>, <a class="txt" href="jordan12.html#E75:41"><i><font color="Green" title="E104">A76</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_1"><i><font color="Green" title="E106">A78</font></i></a>, <a class="txt" href="jordan12.html#E8:41_15_1"><i><font color="Green" title="E109">A81</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_1_1"><i><font color="Green" title="E113">A85</font></i></a>, <a class="ref" href="jordan12.html#T30" target="_self" title="JORDAN12:th.30">Th30</a>, <a class="ref" href="jordan1g.html#T7" title="JORDAN1G:th.7">JORDAN1G:7</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E113">A86</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_15_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_15_1_2"/><i><font color="Green" title="E114">A87</font></i>: 
( <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c10">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c2">f</font></span>)</span> )
 ;<br/><a NAME="E2:41_15_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c13">rl'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>,1</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> <font color="Olive" title="b1">r</font> &amp; <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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c11">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">s</font> &amp; <font color="Olive" title="b2">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c11">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan12.html#E17:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan12.html#E18:41"><i><font color="Green" title="E53">A16</font></i></a>, <a class="txt" href="jordan12.html#E31:41"><i><font color="Green" title="E60">A23</font></i></a>, <a class="txt" href="jordan12.html#E32:41"><i><font color="Green" title="E61">A24</font></i></a>, <a class="txt" href="jordan12.html#E33:41"><i><font color="Green" title="E62">A25</font></i></a>, <a class="txt" href="jordan12.html#E72:41"><i><font color="Green" title="E101">A72</font></i></a>, <a class="txt" href="jordan12.html#E76:41"><i><font color="Green" title="E105">A77</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_1"><i><font color="Green" title="E106">A78</font></i></a>, <a class="txt" href="jordan12.html#E5:41_15_1"><i><font color="Green" title="E108">A80</font></i></a>, <a class="ref" href="jordan12.html#T30" target="_self" title="JORDAN12:th.30">Th30</a>, <a class="ref" href="jordan1g.html#T7" title="JORDAN1G:th.7">JORDAN1G:7</a></i>;<br/><a NAME="E3:41_15_1_2"/><b>then </b>
( <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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>,1</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> <font color="Maroon" title="c14">rl</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c11">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c11">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan12.html#E17:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan12.html#E18:41"><i><font color="Green" title="E53">A16</font></i></a>, <a class="txt" href="jordan12.html#E31:41"><i><font color="Green" title="E60">A23</font></i></a>, <a class="txt" href="jordan12.html#E32:41"><i><font color="Green" title="E61">A24</font></i></a>, <a class="txt" href="jordan12.html#E33:41"><i><font color="Green" title="E62">A25</font></i></a>, <a class="txt" href="jordan12.html#E49:41"><i><font color="Green" title="E78">A41</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_1"><i><font color="Green" title="E106">A78</font></i></a>, <a class="txt" href="jordan12.html#E8:41_15_1"><i><font color="Green" title="E109">A81</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_1_2"><i><font color="Green" title="E114">A87</font></i></a>, <a class="ref" href="jordan12.html#T30" target="_self" title="JORDAN12:th.30">Th30</a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a>, <a class="ref" href="jordan1g.html#T7" title="JORDAN1G:th.7">JORDAN1G:7</a></i>;<br/><b>hence </b><a NAME="E4:41_15_1_2"/>
<font color="Maroon" title="c14">rl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E17:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan12.html#E18:41"><i><font color="Green" title="E53">A16</font></i></a>, <a class="txt" href="jordan12.html#E31:41"><i><font color="Green" title="E60">A23</font></i></a>, <a class="txt" href="jordan12.html#E32:41"><i><font color="Green" title="E61">A24</font></i></a>, <a class="txt" href="jordan12.html#E33:41"><i><font color="Green" title="E62">A25</font></i></a>, <a class="txt" href="jordan12.html#E72:41"><i><font color="Green" title="E101">A72</font></i></a>, <a class="txt" href="jordan12.html#E76:41"><i><font color="Green" title="E105">A77</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_1"><i><font color="Green" title="E106">A78</font></i></a>, <a class="txt" href="jordan12.html#E8:41_15_1"><i><font color="Green" title="E109">A81</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_1_2"><i><font color="Green" title="E114">A87</font></i></a>, <a class="ref" href="jordan12.html#T30" target="_self" title="JORDAN12:th.30">Th30</a>, <a class="ref" href="jordan1g.html#T7" title="JORDAN1G:th.7">JORDAN1G:7</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E114">A88</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_15_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_15_1_3"/><i><font color="Green" title="E115">A89</font></i>: 
( <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> )
 ;<br/><a NAME="E2:41_15_1_3"/><b>then </b>
 ex <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c16">rp'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c11">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">s</font> &amp; <font color="Olive" title="b2">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c11">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan12.html#E17:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan12.html#E18:41"><i><font color="Green" title="E53">A16</font></i></a>, <a class="txt" href="jordan12.html#E31:41"><i><font color="Green" title="E60">A23</font></i></a>, <a class="txt" href="jordan12.html#E32:41"><i><font color="Green" title="E61">A24</font></i></a>, <a class="txt" href="jordan12.html#E33:41"><i><font color="Green" title="E62">A25</font></i></a>, <a class="txt" href="jordan12.html#E72:41"><i><font color="Green" title="E101">A72</font></i></a>, <a class="txt" href="jordan12.html#E74:41"><i><font color="Green" title="E103">A74</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_1"><i><font color="Green" title="E106">A78</font></i></a>, <a class="txt" href="jordan12.html#E11:41_15_1"><i><font color="Green" title="E110">A82</font></i></a>, <a class="ref" href="jordan12.html#T30" target="_self" title="JORDAN12:th.30">Th30</a>, <a class="ref" href="jordan1g.html#T7" title="JORDAN1G:th.7">JORDAN1G:7</a></i>;<br/><a NAME="E3:41_15_1_3"/><b>then </b>
<font color="Maroon" title="c16">rp'</font> <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"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/><b>hence </b><a NAME="E4:41_15_1_3"/>
<font color="Maroon" title="c17">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E17:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan12.html#E18:41"><i><font color="Green" title="E53">A16</font></i></a>, <a class="txt" href="jordan12.html#E31:41"><i><font color="Green" title="E60">A23</font></i></a>, <a class="txt" href="jordan12.html#E32:41"><i><font color="Green" title="E61">A24</font></i></a>, <a class="txt" href="jordan12.html#E33:41"><i><font color="Green" title="E62">A25</font></i></a>, <a class="txt" href="jordan12.html#E49:41"><i><font color="Green" title="E78">A41</font></i></a>, <a class="txt" href="jordan12.html#E72:41"><i><font color="Green" title="E101">A72</font></i></a>, <a class="txt" href="jordan12.html#E74:41"><i><font color="Green" title="E103">A74</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_1"><i><font color="Green" title="E106">A78</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_1_3"><i><font color="Green" title="E115">A89</font></i></a>, <a class="ref" href="jordan12.html#T30" target="_self" title="JORDAN12:th.30">Th30</a>, <a class="ref" href="jordan1g.html#T7" title="JORDAN1G:th.7">JORDAN1G:7</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E115">A90</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_15_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_15_1_4"/><i><font color="Green" title="E116">A91</font></i>: 
( <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">i2</font> )
 ;<br/><a NAME="E2:41_15_1_4"/><b>then </b>
 ex <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c16">rp'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,1</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> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c11">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">s</font> &amp; <font color="Olive" title="b2">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c11">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan12.html#E17:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan12.html#E18:41"><i><font color="Green" title="E53">A16</font></i></a>, <a class="txt" href="jordan12.html#E31:41"><i><font color="Green" title="E60">A23</font></i></a>, <a class="txt" href="jordan12.html#E32:41"><i><font color="Green" title="E61">A24</font></i></a>, <a class="txt" href="jordan12.html#E33:41"><i><font color="Green" title="E62">A25</font></i></a>, <a class="txt" href="jordan12.html#E72:41"><i><font color="Green" title="E101">A72</font></i></a>, <a class="txt" href="jordan12.html#E73:41"><i><font color="Green" title="E102">A73</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_1"><i><font color="Green" title="E106">A78</font></i></a>, <a class="txt" href="jordan12.html#E11:41_15_1"><i><font color="Green" title="E110">A82</font></i></a>, <a class="ref" href="jordan12.html#T30" target="_self" title="JORDAN12:th.30">Th30</a>, <a class="ref" href="jordan1g.html#T7" title="JORDAN1G:th.7">JORDAN1G:7</a></i>;<br/><a NAME="E3:41_15_1_4"/><b>then </b>
( <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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,1</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> <font color="Maroon" title="c16">rp'</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c16">rp'</font> <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"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/><b>hence </b><a NAME="E4:41_15_1_4"/>
<font color="Maroon" title="c17">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E17:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan12.html#E18:41"><i><font color="Green" title="E53">A16</font></i></a>, <a class="txt" href="jordan12.html#E31:41"><i><font color="Green" title="E60">A23</font></i></a>, <a class="txt" href="jordan12.html#E32:41"><i><font color="Green" title="E61">A24</font></i></a>, <a class="txt" href="jordan12.html#E33:41"><i><font color="Green" title="E62">A25</font></i></a>, <a class="txt" href="jordan12.html#E49:41"><i><font color="Green" title="E78">A41</font></i></a>, <a class="txt" href="jordan12.html#E72:41"><i><font color="Green" title="E101">A72</font></i></a>, <a class="txt" href="jordan12.html#E73:41"><i><font color="Green" title="E102">A73</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_1"><i><font color="Green" title="E106">A78</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_1_4"><i><font color="Green" title="E116">A91</font></i></a>, <a class="ref" href="jordan12.html#T30" target="_self" title="JORDAN12:th.30">Th30</a>, <a class="ref" href="jordan1g.html#T7" title="JORDAN1G:th.7">JORDAN1G:7</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E116">A92</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_15_1_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_15_1_5"/><i><font color="Green" title="E117">A93</font></i>: 
( <font color="Maroon" title="c11">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c8">i1</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#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span> )
 ;<br/><a NAME="E2:41_15_1_5"/><b>then </b>
 ex <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c16">rp'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">i1</font>,1</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> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">s</font> &amp; <font color="Olive" title="b2">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c9">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan12.html#E17:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan12.html#E18:41"><i><font color="Green" title="E53">A16</font></i></a>, <a class="txt" href="jordan12.html#E31:41"><i><font color="Green" title="E60">A23</font></i></a>, <a class="txt" href="jordan12.html#E32:41"><i><font color="Green" title="E61">A24</font></i></a>, <a class="txt" href="jordan12.html#E33:41"><i><font color="Green" title="E62">A25</font></i></a>, <a class="txt" href="jordan12.html#E66:41"><i><font color="Green" title="E95">A65</font></i></a>, <a class="txt" href="jordan12.html#E69:41"><i><font color="Green" title="E98">A69</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_1"><i><font color="Green" title="E106">A78</font></i></a>, <a class="txt" href="jordan12.html#E11:41_15_1"><i><font color="Green" title="E110">A82</font></i></a>, <a class="ref" href="jordan12.html#T30" target="_self" title="JORDAN12:th.30">Th30</a>, <a class="ref" href="jordan1g.html#T7" title="JORDAN1G:th.7">JORDAN1G:7</a></i>;<br/><a NAME="E3:41_15_1_5"/><b>then </b>
( <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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">i1</font>,1</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> <font color="Maroon" title="c16">rp'</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c16">rp'</font> <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"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/><b>hence </b><a NAME="E4:41_15_1_5"/>
<font color="Maroon" title="c17">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E17:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan12.html#E18:41"><i><font color="Green" title="E53">A16</font></i></a>, <a class="txt" href="jordan12.html#E31:41"><i><font color="Green" title="E60">A23</font></i></a>, <a class="txt" href="jordan12.html#E32:41"><i><font color="Green" title="E61">A24</font></i></a>, <a class="txt" href="jordan12.html#E33:41"><i><font color="Green" title="E62">A25</font></i></a>, <a class="txt" href="jordan12.html#E50:41"><i><font color="Green" title="E79">A44</font></i></a>, <a class="txt" href="jordan12.html#E66:41"><i><font color="Green" title="E95">A65</font></i></a>, <a class="txt" href="jordan12.html#E69:41"><i><font color="Green" title="E98">A69</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_1"><i><font color="Green" title="E106">A78</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_1_5"><i><font color="Green" title="E117">A93</font></i></a>, <a class="ref" href="jordan12.html#T30" target="_self" title="JORDAN12:th.30">Th30</a>, <a class="ref" href="jordan1g.html#T7" title="JORDAN1G:th.7">JORDAN1G:7</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E117">A94</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_15_1_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_15_1_6"/><i><font color="Green" title="E118">A95</font></i>: 
( <font color="Maroon" title="c11">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c2">f</font></span>)</span> )
 ;<br/><a NAME="E2:41_15_1_6"/><b>then </b>
 ex <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c16">rp'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>,1</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> <font color="Olive" title="b1">r</font> &amp; <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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">s</font> &amp; <font color="Olive" title="b2">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c9">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan12.html#E17:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan12.html#E18:41"><i><font color="Green" title="E53">A16</font></i></a>, <a class="txt" href="jordan12.html#E31:41"><i><font color="Green" title="E60">A23</font></i></a>, <a class="txt" href="jordan12.html#E32:41"><i><font color="Green" title="E61">A24</font></i></a>, <a class="txt" href="jordan12.html#E33:41"><i><font color="Green" title="E62">A25</font></i></a>, <a class="txt" href="jordan12.html#E66:41"><i><font color="Green" title="E95">A65</font></i></a>, <a class="txt" href="jordan12.html#E70:41"><i><font color="Green" title="E99">A70</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_1"><i><font color="Green" title="E106">A78</font></i></a>, <a class="txt" href="jordan12.html#E11:41_15_1"><i><font color="Green" title="E110">A82</font></i></a>, <a class="ref" href="jordan12.html#T30" target="_self" title="JORDAN12:th.30">Th30</a>, <a class="ref" href="jordan1g.html#T7" title="JORDAN1G:th.7">JORDAN1G:7</a></i>;<br/><a NAME="E3:41_15_1_6"/><b>then </b>
<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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>,1</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> <font color="Maroon" title="c16">rp'</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/><b>hence </b><a NAME="E4:41_15_1_6"/>
<font color="Maroon" title="c17">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E17:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan12.html#E18:41"><i><font color="Green" title="E53">A16</font></i></a>, <a class="txt" href="jordan12.html#E31:41"><i><font color="Green" title="E60">A23</font></i></a>, <a class="txt" href="jordan12.html#E32:41"><i><font color="Green" title="E61">A24</font></i></a>, <a class="txt" href="jordan12.html#E33:41"><i><font color="Green" title="E62">A25</font></i></a>, <a class="txt" href="jordan12.html#E50:41"><i><font color="Green" title="E79">A44</font></i></a>, <a class="txt" href="jordan12.html#E66:41"><i><font color="Green" title="E95">A65</font></i></a>, <a class="txt" href="jordan12.html#E70:41"><i><font color="Green" title="E99">A70</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_1"><i><font color="Green" title="E106">A78</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_1_6"><i><font color="Green" title="E118">A95</font></i></a>, <a class="ref" href="jordan12.html#T30" target="_self" title="JORDAN12:th.30">Th30</a>, <a class="ref" href="jordan1g.html#T7" title="JORDAN1G:th.7">JORDAN1G:7</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E118">A96</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_15_1_7"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_15_1_7"/><i><font color="Green" title="E119">A97</font></i>: 
( <font color="Maroon" title="c11">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i1</font> )
 ;<br/><a NAME="E2:41_15_1_7"/><b>then </b>
 ex <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c13">rl'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">s</font> &amp; <font color="Olive" title="b2">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c9">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan12.html#E17:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan12.html#E18:41"><i><font color="Green" title="E53">A16</font></i></a>, <a class="txt" href="jordan12.html#E31:41"><i><font color="Green" title="E60">A23</font></i></a>, <a class="txt" href="jordan12.html#E32:41"><i><font color="Green" title="E61">A24</font></i></a>, <a class="txt" href="jordan12.html#E33:41"><i><font color="Green" title="E62">A25</font></i></a>, <a class="txt" href="jordan12.html#E66:41"><i><font color="Green" title="E95">A65</font></i></a>, <a class="txt" href="jordan12.html#E68:41"><i><font color="Green" title="E97">A67</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_1"><i><font color="Green" title="E106">A78</font></i></a>, <a class="txt" href="jordan12.html#E5:41_15_1"><i><font color="Green" title="E108">A80</font></i></a>, <a class="ref" href="jordan12.html#T30" target="_self" title="JORDAN12:th.30">Th30</a>, <a class="ref" href="jordan1g.html#T7" title="JORDAN1G:th.7">JORDAN1G:7</a></i>;<br/><a NAME="E3:41_15_1_7"/><b>then </b>
<font color="Maroon" title="c13">rl'</font> <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"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/><b>hence </b><a NAME="E4:41_15_1_7"/>
<font color="Maroon" title="c14">rl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E17:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan12.html#E18:41"><i><font color="Green" title="E53">A16</font></i></a>, <a class="txt" href="jordan12.html#E31:41"><i><font color="Green" title="E60">A23</font></i></a>, <a class="txt" href="jordan12.html#E32:41"><i><font color="Green" title="E61">A24</font></i></a>, <a class="txt" href="jordan12.html#E33:41"><i><font color="Green" title="E62">A25</font></i></a>, <a class="txt" href="jordan12.html#E50:41"><i><font color="Green" title="E79">A44</font></i></a>, <a class="txt" href="jordan12.html#E66:41"><i><font color="Green" title="E95">A65</font></i></a>, <a class="txt" href="jordan12.html#E68:41"><i><font color="Green" title="E97">A67</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_1"><i><font color="Green" title="E106">A78</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_1_7"><i><font color="Green" title="E119">A97</font></i></a>, <a class="ref" href="jordan12.html#T30" target="_self" title="JORDAN12:th.30">Th30</a>, <a class="ref" href="jordan1g.html#T7" title="JORDAN1G:th.7">JORDAN1G:7</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E119">A98</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_15_1_8"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_15_1_8"/><i><font color="Green" title="E120">A99</font></i>: 
( <font color="Maroon" title="c11">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">i1</font> )
 ;<br/><a NAME="E2:41_15_1_8"/><b>then </b>
 ex <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c13">rl'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,1</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> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">s</font> &amp; <font color="Olive" title="b2">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c9">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan12.html#E17:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan12.html#E18:41"><i><font color="Green" title="E53">A16</font></i></a>, <a class="txt" href="jordan12.html#E31:41"><i><font color="Green" title="E60">A23</font></i></a>, <a class="txt" href="jordan12.html#E32:41"><i><font color="Green" title="E61">A24</font></i></a>, <a class="txt" href="jordan12.html#E33:41"><i><font color="Green" title="E62">A25</font></i></a>, <a class="txt" href="jordan12.html#E66:41"><i><font color="Green" title="E95">A65</font></i></a>, <a class="txt" href="jordan12.html#E67:41"><i><font color="Green" title="E96">A66</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_1"><i><font color="Green" title="E106">A78</font></i></a>, <a class="txt" href="jordan12.html#E5:41_15_1"><i><font color="Green" title="E108">A80</font></i></a>, <a class="ref" href="jordan12.html#T30" target="_self" title="JORDAN12:th.30">Th30</a>, <a class="ref" href="jordan1g.html#T7" title="JORDAN1G:th.7">JORDAN1G:7</a></i>;<br/><a NAME="E3:41_15_1_8"/><b>then </b>
( <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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,1</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> <font color="Maroon" title="c13">rl'</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c13">rl'</font> <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"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/><b>hence </b><a NAME="E4:41_15_1_8"/>
<font color="Maroon" title="c14">rl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E17:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan12.html#E18:41"><i><font color="Green" title="E53">A16</font></i></a>, <a class="txt" href="jordan12.html#E31:41"><i><font color="Green" title="E60">A23</font></i></a>, <a class="txt" href="jordan12.html#E32:41"><i><font color="Green" title="E61">A24</font></i></a>, <a class="txt" href="jordan12.html#E33:41"><i><font color="Green" title="E62">A25</font></i></a>, <a class="txt" href="jordan12.html#E50:41"><i><font color="Green" title="E79">A44</font></i></a>, <a class="txt" href="jordan12.html#E66:41"><i><font color="Green" title="E95">A65</font></i></a>, <a class="txt" href="jordan12.html#E67:41"><i><font color="Green" title="E96">A66</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_1"><i><font color="Green" title="E106">A78</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_1_8"><i><font color="Green" title="E120">A99</font></i></a>, <a class="ref" href="jordan12.html#T30" target="_self" title="JORDAN12:th.30">Th30</a>, <a class="ref" href="jordan1g.html#T7" title="JORDAN1G:th.7">JORDAN1G:7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_15_1_9"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_1_9_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_15_1_9_1"/>
( <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c11">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span><b>by </b><i><a class="txt" href="jordan12.html#E19:41"><i><font color="Green" title="E54">A17</font></i></a>, <a class="txt" href="jordan12.html#E34:41"><i><font color="Green" title="E63">A26</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_1"><i><font color="Green" title="E106">A78</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_1_9_1_1"><b>suppose </b></a><a NAME="E1:41_15_1_9_1_1"/><i><font color="Green" title="E120">A100</font></i>: 
<font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><div class="add"><a NAME="E2:41_15_1_9_1_1"/><i><font color="Green" title="E121">A101</font></i>: 
<font color="Maroon" title="c14">rl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_15_1_9_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_1_9_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_15_1_9_1_1_1_1"/>
( <font color="Maroon" title="c10">i2</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#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span> or <font color="Maroon" title="c10">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c2">f</font></span>)</span> )
 </span><b>by </b><i><a class="txt" href="jordan12.html#E36:41"><i><font color="Green" title="E65">A28</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_1_9_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:41_15_1_9_1_1_1_1_1"/>
<font color="Maroon" title="c10">i2</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#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_15_1_9_1_1_1_1_1"/>
<font color="Maroon" title="c14">rl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E15:41_15_1"><i><font color="Green" title="E112">A84</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_1_9_1_1"><i><font color="Green" title="E120">A100</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_1_9_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:41_15_1_9_1_1_1_1_2"/>
<font color="Maroon" title="c10">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c2">f</font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_15_1_9_1_1_1_1_2"/>
<font color="Maroon" title="c14">rl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E16:41_15_1"><i><font color="Green" title="E113">A86</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_1_9_1_1"><i><font color="Green" title="E120">A100</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:41_15_1_9_1_1"/><i><font color="Green" title="E122">A102</font></i>: 
<font color="Maroon" title="c17">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_15_1_9_1_1_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_1_9_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_15_1_9_1_1_2_1"/>
( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">i2</font> or 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> )
 </span><b>by </b><i><a class="txt" href="jordan12.html#E37:41"><i><font color="Green" title="E66">A29</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_1_9_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:41_15_1_9_1_1_2_1_1"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">i2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_15_1_9_1_1_2_1_1"/>
<font color="Maroon" title="c17">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E18:41_15_1"><i><font color="Green" title="E115">A90</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_1_9_1_1"><i><font color="Green" title="E120">A100</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_1_9_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:41_15_1_9_1_1_2_1_2"/>
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_15_1_9_1_1_2_1_2"/>
<font color="Maroon" title="c17">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E17:41_15_1"><i><font color="Green" title="E114">A88</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_1_9_1_1"><i><font color="Green" title="E120">A100</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E4:41_15_1_9_1_1"/><i><font color="Green" title="E123">A103</font></i>: 
<font color="Maroon" title="c14">rl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E15:41"><i><font color="Green" title="E51">A13</font></i></a>, <a class="txt" href="jordan12.html#E2:41_15_1_9_1_1"><i><font color="Green" title="E121">A101</font></i></a></i>;<br/><a NAME="E5:41_15_1_9_1_1"/><i><font color="Green" title="E124">A104</font></i>: 
<font color="Maroon" title="c17">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E14:41"><i><font color="Green" title="E50">A11</font></i></a>, <a class="txt" href="jordan12.html#E3:41_15_1_9_1_1"><i><font color="Green" title="E122">A102</font></i></a></i>;<br/><a NAME="E6:41_15_1_9_1_1"/><i><font color="Green" title="E125">A105</font></i>: 
not <font color="Maroon" title="c14">rl</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="c2">f</font>
 <b>by </b><i><a class="txt" href="jordan12.html#E15:41"><i><font color="Green" title="E51">A13</font></i></a>, <a class="txt" href="jordan12.html#E2:41_15_1_9_1_1"><i><font color="Green" title="E121">A101</font></i></a></i>;<br/><a NAME="E7:41_15_1_9_1_1"/>
not <font color="Maroon" title="c17">rp</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="c2">f</font>
 <b>by </b><i><a class="txt" href="jordan12.html#E14:41"><i><font color="Green" title="E50">A11</font></i></a>, <a class="txt" href="jordan12.html#E3:41_15_1_9_1_1"><i><font color="Green" title="E122">A102</font></i></a></i>;<br/><b>hence </b><a NAME="E8:41_15_1_9_1_1"/>
for <font color="Olive" title="b1">C</font> being  <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>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c3">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c4">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 <b>by </b><i><a class="txt" href="jordan12.html#E1:41"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="jordan12.html#E3:41"><i><font color="Green" title="E42">A3</font></i></a>, <a class="txt" href="jordan12.html#E11:41"><i><font color="Green" title="E48">A9</font></i></a>, <a class="txt" href="jordan12.html#E13:41"><i><font color="Green" title="E49">A10</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_1"><i><font color="Green" title="E106">A78</font></i></a>, <a class="txt" href="jordan12.html#E2:41_15_1"><i><font color="Green" title="E107">A79</font></i></a>, <a class="txt" href="jordan12.html#E8:41_15_1"><i><font color="Green" title="E109">A81</font></i></a>, <a class="txt" href="jordan12.html#E14:41_15_1"><i><font color="Green" title="E111">A83</font></i></a>, <a class="txt" href="jordan12.html#E4:41_15_1_9_1_1"><i><font color="Green" title="E123">A103</font></i></a>, <a class="txt" href="jordan12.html#E5:41_15_1_9_1_1"><i><font color="Green" title="E124">A104</font></i></a>, <a class="txt" href="jordan12.html#E6:41_15_1_9_1_1"><i><font color="Green" title="E125">A105</font></i></a>, <a class="ref" href="jordan12.html#T32" target="_self" title="JORDAN12:th.32">Th32</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_1_9_1_2"><b>suppose </b></a><a NAME="E1:41_15_1_9_1_2"/><i><font color="Green" title="E120">A106</font></i>: 
<font color="Maroon" title="c11">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><div class="add"><a NAME="E2:41_15_1_9_1_2"/><i><font color="Green" title="E121">A107</font></i>: 
<font color="Maroon" title="c14">rl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_15_1_9_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_1_9_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_15_1_9_1_2_1_1"/>
( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">i1</font> or 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i1</font> )
 </span><b>by </b><i><a class="txt" href="jordan12.html#E39:41"><i><font color="Green" title="E68">A31</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_1_9_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:41_15_1_9_1_2_1_1_1"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">i1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_15_1_9_1_2_1_1_1"/>
<font color="Maroon" title="c14">rl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E22:41_15_1"><i><font color="Green" title="E119">A98</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_1_9_1_2"><i><font color="Green" title="E120">A106</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_1_9_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:41_15_1_9_1_2_1_1_2"/>
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_15_1_9_1_2_1_1_2"/>
<font color="Maroon" title="c14">rl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E21:41_15_1"><i><font color="Green" title="E118">A96</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_1_9_1_2"><i><font color="Green" title="E120">A106</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:41_15_1_9_1_2"/><i><font color="Green" title="E122">A108</font></i>: 
<font color="Maroon" title="c17">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_15_1_9_1_2_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_1_9_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_15_1_9_1_2_2_1"/>
( <font color="Maroon" title="c8">i1</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#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span> or <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c2">f</font></span>)</span> )
 </span><b>by </b><i><a class="txt" href="jordan12.html#E38:41"><i><font color="Green" title="E67">A30</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_1_9_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:41_15_1_9_1_2_2_1_1"/>
<font color="Maroon" title="c8">i1</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#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_15_1_9_1_2_2_1_1"/>
<font color="Maroon" title="c17">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E19:41_15_1"><i><font color="Green" title="E116">A92</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_1_9_1_2"><i><font color="Green" title="E120">A106</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_1_9_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:41_15_1_9_1_2_2_1_2"/>
<font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c2">f</font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_15_1_9_1_2_2_1_2"/>
<font color="Maroon" title="c17">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E20:41_15_1"><i><font color="Green" title="E117">A94</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_1_9_1_2"><i><font color="Green" title="E120">A106</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E4:41_15_1_9_1_2"/><i><font color="Green" title="E123">A109</font></i>: 
<font color="Maroon" title="c14">rl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E15:41"><i><font color="Green" title="E51">A13</font></i></a>, <a class="txt" href="jordan12.html#E2:41_15_1_9_1_2"><i><font color="Green" title="E121">A107</font></i></a></i>;<br/><a NAME="E5:41_15_1_9_1_2"/><i><font color="Green" title="E124">A110</font></i>: 
<font color="Maroon" title="c17">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E14:41"><i><font color="Green" title="E50">A11</font></i></a>, <a class="txt" href="jordan12.html#E3:41_15_1_9_1_2"><i><font color="Green" title="E122">A108</font></i></a></i>;<br/><a NAME="E6:41_15_1_9_1_2"/><i><font color="Green" title="E125">A111</font></i>: 
not <font color="Maroon" title="c14">rl</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="c2">f</font>
 <b>by </b><i><a class="txt" href="jordan12.html#E15:41"><i><font color="Green" title="E51">A13</font></i></a>, <a class="txt" href="jordan12.html#E2:41_15_1_9_1_2"><i><font color="Green" title="E121">A107</font></i></a></i>;<br/><a NAME="E7:41_15_1_9_1_2"/>
not <font color="Maroon" title="c17">rp</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="c2">f</font>
 <b>by </b><i><a class="txt" href="jordan12.html#E14:41"><i><font color="Green" title="E50">A11</font></i></a>, <a class="txt" href="jordan12.html#E3:41_15_1_9_1_2"><i><font color="Green" title="E122">A108</font></i></a></i>;<br/><b>hence </b><a NAME="E8:41_15_1_9_1_2"/>
for <font color="Olive" title="b1">C</font> being  <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>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c3">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c4">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 <b>by </b><i><a class="txt" href="jordan12.html#E1:41"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="jordan12.html#E3:41"><i><font color="Green" title="E42">A3</font></i></a>, <a class="txt" href="jordan12.html#E11:41"><i><font color="Green" title="E48">A9</font></i></a>, <a class="txt" href="jordan12.html#E13:41"><i><font color="Green" title="E49">A10</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_1"><i><font color="Green" title="E106">A78</font></i></a>, <a class="txt" href="jordan12.html#E2:41_15_1"><i><font color="Green" title="E107">A79</font></i></a>, <a class="txt" href="jordan12.html#E8:41_15_1"><i><font color="Green" title="E109">A81</font></i></a>, <a class="txt" href="jordan12.html#E14:41_15_1"><i><font color="Green" title="E111">A83</font></i></a>, <a class="txt" href="jordan12.html#E4:41_15_1_9_1_2"><i><font color="Green" title="E123">A109</font></i></a>, <a class="txt" href="jordan12.html#E5:41_15_1_9_1_2"><i><font color="Green" title="E124">A110</font></i></a>, <a class="txt" href="jordan12.html#E6:41_15_1_9_1_2"><i><font color="Green" title="E125">A111</font></i></a>, <a class="ref" href="jordan12.html#T32" target="_self" title="JORDAN12:th.32">Th32</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E24:41_15_1"/>
for <font color="Olive" title="b1">C</font> being  <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>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c3">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c4">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_2"><b>suppose </b></a><a NAME="E1:41_15_2"/><i><font color="Green" title="E106">A112</font></i>: 
<font color="Maroon" title="c3">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 ;<br/><div class="add"><a NAME="E2:41_15_2"/><b>then </b><i><font color="Green" title="E107">A113</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="jordan12.html#E7:41"><i><font color="Green" title="E46">A7</font></i></a>, <a class="ref" href="goboard7.html#T5" title="GOBOARD7:th.5">GOBOARD7:5</a></i>;<br/><a NAME="E3:41_15_2"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="jordan12.html#E8:41"><i><font color="Green" title="E47">A8</font></i></a>, <a class="txt" href="jordan12.html#E13:41"><i><font color="Green" title="E49">A10</font></i></a>, <a class="ref" href="jordan9.html#T11" title="JORDAN9:th.11">JORDAN9:11</a></i>;<br/><b>then consider </b><font color="Maroon" title="c12">rl'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:41_15_2"/><i><font color="Green" title="E108">A114</font></i>: 
<font color="Maroon" title="c12">rl'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 <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="c13">rl'</font> = <font color="Maroon" title="c12">rl'</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="jordan12.html#E5:41_15_2"><i><font color="Green" title="E108">A114</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c14">rl</font> = <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">rl'</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> 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> ;<br/><a NAME="E8:41_15_2"/><i><font color="Green" title="E109">A115</font></i>: 
( <font color="Maroon" title="c14">rl</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c14">rl</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">rl'</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/><a NAME="E9:41_15_2"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="jordan12.html#E8:41"><i><font color="Green" title="E47">A8</font></i></a>, <a class="txt" href="jordan12.html#E13:41"><i><font color="Green" title="E49">A10</font></i></a>, <a class="ref" href="jordan9.html#T11" title="JORDAN9:th.11">JORDAN9:11</a></i>;<br/><b>then consider </b><font color="Maroon" title="c15">rp'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E11:41_15_2"/><i><font color="Green" title="E110">A116</font></i>: 
<font color="Maroon" title="c15">rp'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 <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="c16">rp'</font> = <font color="Maroon" title="c15">rp'</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="jordan12.html#E11:41_15_2"><i><font color="Green" title="E110">A116</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c17">rp</font> = <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">rp'</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> 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> ;<br/><a NAME="E14:41_15_2"/><i><font color="Green" title="E111">A117</font></i>: 
( <font color="Maroon" title="c17">rp</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c17">rp</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">rp'</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/><div><i><font color="Green" title="E112">A118</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_15_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_15_2_1"/><i><font color="Green" title="E113">A119</font></i>: 
( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c9">j1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="c2">f</font></span>)</span> )
 ;<br/><a NAME="E2:41_15_2_1"/><b>then </b>
 ex <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c16">rp'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i2</font>,1</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> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">s</font> &amp; <font color="Olive" title="b2">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c9">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan12.html#E17:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan12.html#E18:41"><i><font color="Green" title="E53">A16</font></i></a>, <a class="txt" href="jordan12.html#E31:41"><i><font color="Green" title="E60">A23</font></i></a>, <a class="txt" href="jordan12.html#E32:41"><i><font color="Green" title="E61">A24</font></i></a>, <a class="txt" href="jordan12.html#E33:41"><i><font color="Green" title="E62">A25</font></i></a>, <a class="txt" href="jordan12.html#E60:41"><i><font color="Green" title="E89">A58</font></i></a>, <a class="txt" href="jordan12.html#E63:41"><i><font color="Green" title="E92">A62</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_2"><i><font color="Green" title="E106">A112</font></i></a>, <a class="txt" href="jordan12.html#E11:41_15_2"><i><font color="Green" title="E110">A116</font></i></a>, <a class="ref" href="jordan12.html#T30" target="_self" title="JORDAN12:th.30">Th30</a>, <a class="ref" href="jordan1g.html#T6" title="JORDAN1G:th.6">JORDAN1G:6</a></i>;<br/><a NAME="E3:41_15_2_1"/><b>then </b>
( <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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">rp</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c17">rp</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c9">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan12.html#E14:41_15_2"><i><font color="Green" title="E111">A117</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/><b>hence </b><a NAME="E4:41_15_2_1"/>
<font color="Maroon" title="c17">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E17:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan12.html#E18:41"><i><font color="Green" title="E53">A16</font></i></a>, <a class="txt" href="jordan12.html#E31:41"><i><font color="Green" title="E60">A23</font></i></a>, <a class="txt" href="jordan12.html#E32:41"><i><font color="Green" title="E61">A24</font></i></a>, <a class="txt" href="jordan12.html#E33:41"><i><font color="Green" title="E62">A25</font></i></a>, <a class="txt" href="jordan12.html#E51:41"><i><font color="Green" title="E80">A46</font></i></a>, <a class="txt" href="jordan12.html#E60:41"><i><font color="Green" title="E89">A58</font></i></a>, <a class="txt" href="jordan12.html#E63:41"><i><font color="Green" title="E92">A62</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_2"><i><font color="Green" title="E106">A112</font></i></a>, <a class="txt" href="jordan12.html#E14:41_15_2"><i><font color="Green" title="E111">A117</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_2_1"><i><font color="Green" title="E113">A119</font></i></a>, <a class="ref" href="jordan12.html#T30" target="_self" title="JORDAN12:th.30">Th30</a>, <a class="ref" href="jordan1g.html#T6" title="JORDAN1G:th.6">JORDAN1G:6</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E113">A120</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_15_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_15_2_2"/><i><font color="Green" title="E114">A121</font></i>: 
( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c9">j1</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#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span> )
 ;<br/><a NAME="E2:41_15_2_2"/><b>then </b>
 ex <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c16">rp'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i2</font>,1</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> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <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="c2">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#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">s</font> )
 <b>by </b><i><a class="txt" href="jordan12.html#E17:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan12.html#E18:41"><i><font color="Green" title="E53">A16</font></i></a>, <a class="txt" href="jordan12.html#E31:41"><i><font color="Green" title="E60">A23</font></i></a>, <a class="txt" href="jordan12.html#E32:41"><i><font color="Green" title="E61">A24</font></i></a>, <a class="txt" href="jordan12.html#E33:41"><i><font color="Green" title="E62">A25</font></i></a>, <a class="txt" href="jordan12.html#E60:41"><i><font color="Green" title="E89">A58</font></i></a>, <a class="txt" href="jordan12.html#E64:41"><i><font color="Green" title="E93">A63</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_2"><i><font color="Green" title="E106">A112</font></i></a>, <a class="txt" href="jordan12.html#E11:41_15_2"><i><font color="Green" title="E110">A116</font></i></a>, <a class="ref" href="jordan12.html#T30" target="_self" title="JORDAN12:th.30">Th30</a>, <a class="ref" href="jordan1g.html#T6" title="JORDAN1G:th.6">JORDAN1G:6</a></i>;<br/><a NAME="E3:41_15_2_2"/><b>then </b>
( <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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i2</font>,1</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> <font color="Maroon" title="c1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c1">p</font> <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"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <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="c2">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#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">rp</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan12.html#E17:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan12.html#E18:41"><i><font color="Green" title="E53">A16</font></i></a>, <a class="txt" href="jordan12.html#E31:41"><i><font color="Green" title="E60">A23</font></i></a>, <a class="txt" href="jordan12.html#E32:41"><i><font color="Green" title="E61">A24</font></i></a>, <a class="txt" href="jordan12.html#E33:41"><i><font color="Green" title="E62">A25</font></i></a>, <a class="txt" href="jordan12.html#E51:41"><i><font color="Green" title="E80">A46</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_2"><i><font color="Green" title="E106">A112</font></i></a>, <a class="txt" href="jordan12.html#E14:41_15_2"><i><font color="Green" title="E111">A117</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_2_2"><i><font color="Green" title="E114">A121</font></i></a>, <a class="ref" href="jordan12.html#T30" target="_self" title="JORDAN12:th.30">Th30</a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a>, <a class="ref" href="jordan1g.html#T6" title="JORDAN1G:th.6">JORDAN1G:6</a></i>;<br/><b>hence </b><a NAME="E4:41_15_2_2"/>
<font color="Maroon" title="c17">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E17:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan12.html#E18:41"><i><font color="Green" title="E53">A16</font></i></a>, <a class="txt" href="jordan12.html#E31:41"><i><font color="Green" title="E60">A23</font></i></a>, <a class="txt" href="jordan12.html#E32:41"><i><font color="Green" title="E61">A24</font></i></a>, <a class="txt" href="jordan12.html#E33:41"><i><font color="Green" title="E62">A25</font></i></a>, <a class="txt" href="jordan12.html#E60:41"><i><font color="Green" title="E89">A58</font></i></a>, <a class="txt" href="jordan12.html#E64:41"><i><font color="Green" title="E93">A63</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_2"><i><font color="Green" title="E106">A112</font></i></a>, <a class="txt" href="jordan12.html#E14:41_15_2"><i><font color="Green" title="E111">A117</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_2_2"><i><font color="Green" title="E114">A121</font></i></a>, <a class="ref" href="jordan12.html#T30" target="_self" title="JORDAN12:th.30">Th30</a>, <a class="ref" href="jordan1g.html#T6" title="JORDAN1G:th.6">JORDAN1G:6</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E114">A122</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_15_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_15_2_3"/><i><font color="Green" title="E115">A123</font></i>: 
( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j1</font> )
 ;<br/><a NAME="E2:41_15_2_3"/><b>then </b>
 ex <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c13">rl'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i2</font>,1</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> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b2">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</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="jordan12.html#E17:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan12.html#E18:41"><i><font color="Green" title="E53">A16</font></i></a>, <a class="txt" href="jordan12.html#E31:41"><i><font color="Green" title="E60">A23</font></i></a>, <a class="txt" href="jordan12.html#E32:41"><i><font color="Green" title="E61">A24</font></i></a>, <a class="txt" href="jordan12.html#E33:41"><i><font color="Green" title="E62">A25</font></i></a>, <a class="txt" href="jordan12.html#E60:41"><i><font color="Green" title="E89">A58</font></i></a>, <a class="txt" href="jordan12.html#E62:41"><i><font color="Green" title="E91">A60</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_2"><i><font color="Green" title="E106">A112</font></i></a>, <a class="txt" href="jordan12.html#E5:41_15_2"><i><font color="Green" title="E108">A114</font></i></a>, <a class="ref" href="jordan12.html#T30" target="_self" title="JORDAN12:th.30">Th30</a>, <a class="ref" href="jordan1g.html#T6" title="JORDAN1G:th.6">JORDAN1G:6</a></i>;<br/><a NAME="E3:41_15_2_3"/><b>then </b>
<font color="Maroon" title="c14">rl</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</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="jordan12.html#E8:41_15_2"><i><font color="Green" title="E109">A115</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/><b>hence </b><a NAME="E4:41_15_2_3"/>
<font color="Maroon" title="c14">rl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E17:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan12.html#E18:41"><i><font color="Green" title="E53">A16</font></i></a>, <a class="txt" href="jordan12.html#E31:41"><i><font color="Green" title="E60">A23</font></i></a>, <a class="txt" href="jordan12.html#E32:41"><i><font color="Green" title="E61">A24</font></i></a>, <a class="txt" href="jordan12.html#E33:41"><i><font color="Green" title="E62">A25</font></i></a>, <a class="txt" href="jordan12.html#E51:41"><i><font color="Green" title="E80">A46</font></i></a>, <a class="txt" href="jordan12.html#E60:41"><i><font color="Green" title="E89">A58</font></i></a>, <a class="txt" href="jordan12.html#E62:41"><i><font color="Green" title="E91">A60</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_2"><i><font color="Green" title="E106">A112</font></i></a>, <a class="txt" href="jordan12.html#E8:41_15_2"><i><font color="Green" title="E109">A115</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_2_3"><i><font color="Green" title="E115">A123</font></i></a>, <a class="ref" href="jordan12.html#T30" target="_self" title="JORDAN12:th.30">Th30</a>, <a class="ref" href="jordan1g.html#T6" title="JORDAN1G:th.6">JORDAN1G:6</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E115">A124</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_15_2_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_15_2_4"/><i><font color="Green" title="E116">A125</font></i>: 
( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">j1</font> )
 ;<br/><a NAME="E2:41_15_2_4"/><b>then </b>
 ex <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c13">rl'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i2</font>,1</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> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c9">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">s</font> &amp; <font color="Olive" title="b2">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan12.html#E17:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan12.html#E18:41"><i><font color="Green" title="E53">A16</font></i></a>, <a class="txt" href="jordan12.html#E31:41"><i><font color="Green" title="E60">A23</font></i></a>, <a class="txt" href="jordan12.html#E32:41"><i><font color="Green" title="E61">A24</font></i></a>, <a class="txt" href="jordan12.html#E33:41"><i><font color="Green" title="E62">A25</font></i></a>, <a class="txt" href="jordan12.html#E60:41"><i><font color="Green" title="E89">A58</font></i></a>, <a class="txt" href="jordan12.html#E61:41"><i><font color="Green" title="E90">A59</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_2"><i><font color="Green" title="E106">A112</font></i></a>, <a class="txt" href="jordan12.html#E5:41_15_2"><i><font color="Green" title="E108">A114</font></i></a>, <a class="ref" href="jordan12.html#T30" target="_self" title="JORDAN12:th.30">Th30</a>, <a class="ref" href="jordan1g.html#T6" title="JORDAN1G:th.6">JORDAN1G:6</a></i>;<br/><a NAME="E3:41_15_2_4"/><b>then </b>
( <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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c9">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">rl'</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c13">rl'</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/><b>hence </b><a NAME="E4:41_15_2_4"/>
<font color="Maroon" title="c14">rl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E17:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan12.html#E18:41"><i><font color="Green" title="E53">A16</font></i></a>, <a class="txt" href="jordan12.html#E31:41"><i><font color="Green" title="E60">A23</font></i></a>, <a class="txt" href="jordan12.html#E32:41"><i><font color="Green" title="E61">A24</font></i></a>, <a class="txt" href="jordan12.html#E33:41"><i><font color="Green" title="E62">A25</font></i></a>, <a class="txt" href="jordan12.html#E51:41"><i><font color="Green" title="E80">A46</font></i></a>, <a class="txt" href="jordan12.html#E60:41"><i><font color="Green" title="E89">A58</font></i></a>, <a class="txt" href="jordan12.html#E61:41"><i><font color="Green" title="E90">A59</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_2"><i><font color="Green" title="E106">A112</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_2_4"><i><font color="Green" title="E116">A125</font></i></a>, <a class="ref" href="jordan12.html#T30" target="_self" title="JORDAN12:th.30">Th30</a>, <a class="ref" href="jordan1g.html#T6" title="JORDAN1G:th.6">JORDAN1G:6</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E116">A126</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_15_2_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_15_2_5"/><i><font color="Green" title="E117">A127</font></i>: 
( <font color="Maroon" title="c10">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c9">j1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="c2">f</font></span>)</span> )
 ;<br/><a NAME="E2:41_15_2_5"/><b>then </b>
 ex <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c13">rl'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">i1</font>,1</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> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">s</font> &amp; <font color="Olive" title="b2">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c9">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan12.html#E17:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan12.html#E18:41"><i><font color="Green" title="E53">A16</font></i></a>, <a class="txt" href="jordan12.html#E31:41"><i><font color="Green" title="E60">A23</font></i></a>, <a class="txt" href="jordan12.html#E32:41"><i><font color="Green" title="E61">A24</font></i></a>, <a class="txt" href="jordan12.html#E33:41"><i><font color="Green" title="E62">A25</font></i></a>, <a class="txt" href="jordan12.html#E53:41"><i><font color="Green" title="E82">A50</font></i></a>, <a class="txt" href="jordan12.html#E57:41"><i><font color="Green" title="E86">A55</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_2"><i><font color="Green" title="E106">A112</font></i></a>, <a class="txt" href="jordan12.html#E5:41_15_2"><i><font color="Green" title="E108">A114</font></i></a>, <a class="ref" href="jordan12.html#T30" target="_self" title="JORDAN12:th.30">Th30</a>, <a class="ref" href="jordan1g.html#T6" title="JORDAN1G:th.6">JORDAN1G:6</a></i>;<br/><a NAME="E3:41_15_2_5"/><b>then </b>
( <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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">rl</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c14">rl</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c9">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan12.html#E8:41_15_2"><i><font color="Green" title="E109">A115</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/><b>hence </b><a NAME="E4:41_15_2_5"/>
<font color="Maroon" title="c14">rl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E17:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan12.html#E18:41"><i><font color="Green" title="E53">A16</font></i></a>, <a class="txt" href="jordan12.html#E31:41"><i><font color="Green" title="E60">A23</font></i></a>, <a class="txt" href="jordan12.html#E32:41"><i><font color="Green" title="E61">A24</font></i></a>, <a class="txt" href="jordan12.html#E33:41"><i><font color="Green" title="E62">A25</font></i></a>, <a class="txt" href="jordan12.html#E52:41"><i><font color="Green" title="E81">A48</font></i></a>, <a class="txt" href="jordan12.html#E53:41"><i><font color="Green" title="E82">A50</font></i></a>, <a class="txt" href="jordan12.html#E57:41"><i><font color="Green" title="E86">A55</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_2"><i><font color="Green" title="E106">A112</font></i></a>, <a class="txt" href="jordan12.html#E8:41_15_2"><i><font color="Green" title="E109">A115</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_2_5"><i><font color="Green" title="E117">A127</font></i></a>, <a class="ref" href="jordan12.html#T30" target="_self" title="JORDAN12:th.30">Th30</a>, <a class="ref" href="jordan1g.html#T6" title="JORDAN1G:th.6">JORDAN1G:6</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E117">A128</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_15_2_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_15_2_6"/><i><font color="Green" title="E118">A129</font></i>: 
( <font color="Maroon" title="c10">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c9">j1</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#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span> )
 ;<br/><a NAME="E2:41_15_2_6"/><b>then </b>
 ex <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c13">rl'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">i1</font>,1</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> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <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="c2">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#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">s</font> )
 <b>by </b><i><a class="txt" href="jordan12.html#E17:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan12.html#E18:41"><i><font color="Green" title="E53">A16</font></i></a>, <a class="txt" href="jordan12.html#E31:41"><i><font color="Green" title="E60">A23</font></i></a>, <a class="txt" href="jordan12.html#E32:41"><i><font color="Green" title="E61">A24</font></i></a>, <a class="txt" href="jordan12.html#E33:41"><i><font color="Green" title="E62">A25</font></i></a>, <a class="txt" href="jordan12.html#E53:41"><i><font color="Green" title="E82">A50</font></i></a>, <a class="txt" href="jordan12.html#E58:41"><i><font color="Green" title="E87">A56</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_2"><i><font color="Green" title="E106">A112</font></i></a>, <a class="txt" href="jordan12.html#E5:41_15_2"><i><font color="Green" title="E108">A114</font></i></a>, <a class="ref" href="jordan12.html#T30" target="_self" title="JORDAN12:th.30">Th30</a>, <a class="ref" href="jordan1g.html#T6" title="JORDAN1G:th.6">JORDAN1G:6</a></i>;<br/><a NAME="E3:41_15_2_6"/><b>then </b>
<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="c2">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#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">rl'</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/><b>hence </b><a NAME="E4:41_15_2_6"/>
<font color="Maroon" title="c14">rl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E17:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan12.html#E18:41"><i><font color="Green" title="E53">A16</font></i></a>, <a class="txt" href="jordan12.html#E31:41"><i><font color="Green" title="E60">A23</font></i></a>, <a class="txt" href="jordan12.html#E32:41"><i><font color="Green" title="E61">A24</font></i></a>, <a class="txt" href="jordan12.html#E33:41"><i><font color="Green" title="E62">A25</font></i></a>, <a class="txt" href="jordan12.html#E52:41"><i><font color="Green" title="E81">A48</font></i></a>, <a class="txt" href="jordan12.html#E53:41"><i><font color="Green" title="E82">A50</font></i></a>, <a class="txt" href="jordan12.html#E58:41"><i><font color="Green" title="E87">A56</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_2"><i><font color="Green" title="E106">A112</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_2_6"><i><font color="Green" title="E118">A129</font></i></a>, <a class="ref" href="jordan12.html#T30" target="_self" title="JORDAN12:th.30">Th30</a>, <a class="ref" href="jordan1g.html#T6" title="JORDAN1G:th.6">JORDAN1G:6</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E118">A130</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_15_2_7"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_15_2_7"/><i><font color="Green" title="E119">A131</font></i>: 
( <font color="Maroon" title="c10">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j1</font> )
 ;<br/><a NAME="E2:41_15_2_7"/><b>then </b>
 ex <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c16">rp'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">i1</font>,1</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> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b2">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</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="jordan12.html#E17:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan12.html#E18:41"><i><font color="Green" title="E53">A16</font></i></a>, <a class="txt" href="jordan12.html#E31:41"><i><font color="Green" title="E60">A23</font></i></a>, <a class="txt" href="jordan12.html#E32:41"><i><font color="Green" title="E61">A24</font></i></a>, <a class="txt" href="jordan12.html#E33:41"><i><font color="Green" title="E62">A25</font></i></a>, <a class="txt" href="jordan12.html#E53:41"><i><font color="Green" title="E82">A50</font></i></a>, <a class="txt" href="jordan12.html#E56:41"><i><font color="Green" title="E85">A53</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_2"><i><font color="Green" title="E106">A112</font></i></a>, <a class="txt" href="jordan12.html#E11:41_15_2"><i><font color="Green" title="E110">A116</font></i></a>, <a class="ref" href="jordan12.html#T30" target="_self" title="JORDAN12:th.30">Th30</a>, <a class="ref" href="jordan1g.html#T6" title="JORDAN1G:th.6">JORDAN1G:6</a></i>;<br/><a NAME="E3:41_15_2_7"/><b>then </b>
<font color="Maroon" title="c16">rp'</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</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="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/><b>hence </b><a NAME="E4:41_15_2_7"/>
<font color="Maroon" title="c17">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E17:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan12.html#E18:41"><i><font color="Green" title="E53">A16</font></i></a>, <a class="txt" href="jordan12.html#E31:41"><i><font color="Green" title="E60">A23</font></i></a>, <a class="txt" href="jordan12.html#E32:41"><i><font color="Green" title="E61">A24</font></i></a>, <a class="txt" href="jordan12.html#E33:41"><i><font color="Green" title="E62">A25</font></i></a>, <a class="txt" href="jordan12.html#E52:41"><i><font color="Green" title="E81">A48</font></i></a>, <a class="txt" href="jordan12.html#E53:41"><i><font color="Green" title="E82">A50</font></i></a>, <a class="txt" href="jordan12.html#E56:41"><i><font color="Green" title="E85">A53</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_2"><i><font color="Green" title="E106">A112</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_2_7"><i><font color="Green" title="E119">A131</font></i></a>, <a class="ref" href="jordan12.html#T30" target="_self" title="JORDAN12:th.30">Th30</a>, <a class="ref" href="jordan1g.html#T6" title="JORDAN1G:th.6">JORDAN1G:6</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E119">A132</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_15_2_8"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_15_2_8"/><i><font color="Green" title="E120">A133</font></i>: 
( <font color="Maroon" title="c10">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">j1</font> )
 ;<br/><a NAME="E2:41_15_2_8"/><b>then </b>
 ex <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c16">rp'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">i1</font>,1</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> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c9">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">s</font> &amp; <font color="Olive" title="b2">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan12.html#E17:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan12.html#E18:41"><i><font color="Green" title="E53">A16</font></i></a>, <a class="txt" href="jordan12.html#E31:41"><i><font color="Green" title="E60">A23</font></i></a>, <a class="txt" href="jordan12.html#E32:41"><i><font color="Green" title="E61">A24</font></i></a>, <a class="txt" href="jordan12.html#E33:41"><i><font color="Green" title="E62">A25</font></i></a>, <a class="txt" href="jordan12.html#E53:41"><i><font color="Green" title="E82">A50</font></i></a>, <a class="txt" href="jordan12.html#E55:41"><i><font color="Green" title="E84">A52</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_2"><i><font color="Green" title="E106">A112</font></i></a>, <a class="txt" href="jordan12.html#E11:41_15_2"><i><font color="Green" title="E110">A116</font></i></a>, <a class="ref" href="jordan12.html#T30" target="_self" title="JORDAN12:th.30">Th30</a>, <a class="ref" href="jordan1g.html#T6" title="JORDAN1G:th.6">JORDAN1G:6</a></i>;<br/><a NAME="E3:41_15_2_8"/><b>then </b>
( <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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c9">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">rp'</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c16">rp'</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/><b>hence </b><a NAME="E4:41_15_2_8"/>
<font color="Maroon" title="c17">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E17:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan12.html#E18:41"><i><font color="Green" title="E53">A16</font></i></a>, <a class="txt" href="jordan12.html#E31:41"><i><font color="Green" title="E60">A23</font></i></a>, <a class="txt" href="jordan12.html#E32:41"><i><font color="Green" title="E61">A24</font></i></a>, <a class="txt" href="jordan12.html#E33:41"><i><font color="Green" title="E62">A25</font></i></a>, <a class="txt" href="jordan12.html#E52:41"><i><font color="Green" title="E81">A48</font></i></a>, <a class="txt" href="jordan12.html#E53:41"><i><font color="Green" title="E82">A50</font></i></a>, <a class="txt" href="jordan12.html#E55:41"><i><font color="Green" title="E84">A52</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_2"><i><font color="Green" title="E106">A112</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_2_8"><i><font color="Green" title="E120">A133</font></i></a>, <a class="ref" href="jordan12.html#T30" target="_self" title="JORDAN12:th.30">Th30</a>, <a class="ref" href="jordan1g.html#T6" title="JORDAN1G:th.6">JORDAN1G:6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_15_2_9"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_2_9_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_15_2_9_1"/>
( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c10">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span><b>by </b><i><a class="txt" href="jordan12.html#E19:41"><i><font color="Green" title="E54">A17</font></i></a>, <a class="txt" href="jordan12.html#E35:41"><i><font color="Green" title="E64">A27</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_2"><i><font color="Green" title="E106">A112</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_2_9_1_1"><b>suppose </b></a><a NAME="E1:41_15_2_9_1_1"/><i><font color="Green" title="E120">A134</font></i>: 
<font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><div class="add"><a NAME="E2:41_15_2_9_1_1"/><i><font color="Green" title="E121">A135</font></i>: 
<font color="Maroon" title="c14">rl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_15_2_9_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_2_9_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_15_2_9_1_1_1_1"/>
( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">j1</font> or 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j1</font> )
 </span><b>by </b><i><a class="txt" href="jordan12.html#E40:41"><i><font color="Green" title="E69">A32</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_2_9_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:41_15_2_9_1_1_1_1_1"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">j1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_15_2_9_1_1_1_1_1"/>
<font color="Maroon" title="c14">rl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E18:41_15_2"><i><font color="Green" title="E115">A124</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_2_9_1_1"><i><font color="Green" title="E120">A134</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_2_9_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:41_15_2_9_1_1_1_1_2"/>
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_15_2_9_1_1_1_1_2"/>
<font color="Maroon" title="c14">rl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E17:41_15_2"><i><font color="Green" title="E114">A122</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_2_9_1_1"><i><font color="Green" title="E120">A134</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:41_15_2_9_1_1"/><i><font color="Green" title="E122">A136</font></i>: 
<font color="Maroon" title="c17">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_15_2_9_1_1_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_2_9_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_15_2_9_1_1_2_1"/>
( <font color="Maroon" title="c9">j1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="c2">f</font></span>)</span> or <font color="Maroon" title="c9">j1</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#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span> )
 </span><b>by </b><i><a class="txt" href="jordan12.html#E41:41"><i><font color="Green" title="E70">A33</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_2_9_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:41_15_2_9_1_1_2_1_1"/>
<font color="Maroon" title="c9">j1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="c2">f</font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_15_2_9_1_1_2_1_1"/>
<font color="Maroon" title="c17">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E15:41_15_2"><i><font color="Green" title="E112">A118</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_2_9_1_1"><i><font color="Green" title="E120">A134</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_2_9_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:41_15_2_9_1_1_2_1_2"/>
<font color="Maroon" title="c9">j1</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#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_15_2_9_1_1_2_1_2"/>
<font color="Maroon" title="c17">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E16:41_15_2"><i><font color="Green" title="E113">A120</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_2_9_1_1"><i><font color="Green" title="E120">A134</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E4:41_15_2_9_1_1"/><i><font color="Green" title="E123">A137</font></i>: 
<font color="Maroon" title="c14">rl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E15:41"><i><font color="Green" title="E51">A13</font></i></a>, <a class="txt" href="jordan12.html#E2:41_15_2_9_1_1"><i><font color="Green" title="E121">A135</font></i></a></i>;<br/><a NAME="E5:41_15_2_9_1_1"/><i><font color="Green" title="E124">A138</font></i>: 
<font color="Maroon" title="c17">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E14:41"><i><font color="Green" title="E50">A11</font></i></a>, <a class="txt" href="jordan12.html#E3:41_15_2_9_1_1"><i><font color="Green" title="E122">A136</font></i></a></i>;<br/><a NAME="E6:41_15_2_9_1_1"/><i><font color="Green" title="E125">A139</font></i>: 
not <font color="Maroon" title="c14">rl</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="c2">f</font>
 <b>by </b><i><a class="txt" href="jordan12.html#E15:41"><i><font color="Green" title="E51">A13</font></i></a>, <a class="txt" href="jordan12.html#E2:41_15_2_9_1_1"><i><font color="Green" title="E121">A135</font></i></a></i>;<br/><a NAME="E7:41_15_2_9_1_1"/>
not <font color="Maroon" title="c17">rp</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="c2">f</font>
 <b>by </b><i><a class="txt" href="jordan12.html#E14:41"><i><font color="Green" title="E50">A11</font></i></a>, <a class="txt" href="jordan12.html#E3:41_15_2_9_1_1"><i><font color="Green" title="E122">A136</font></i></a></i>;<br/><b>hence </b><a NAME="E8:41_15_2_9_1_1"/>
for <font color="Olive" title="b1">C</font> being  <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>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c3">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c4">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 <b>by </b><i><a class="txt" href="jordan12.html#E1:41"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="jordan12.html#E3:41"><i><font color="Green" title="E42">A3</font></i></a>, <a class="txt" href="jordan12.html#E11:41"><i><font color="Green" title="E48">A9</font></i></a>, <a class="txt" href="jordan12.html#E13:41"><i><font color="Green" title="E49">A10</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_2"><i><font color="Green" title="E106">A112</font></i></a>, <a class="txt" href="jordan12.html#E2:41_15_2"><i><font color="Green" title="E107">A113</font></i></a>, <a class="txt" href="jordan12.html#E8:41_15_2"><i><font color="Green" title="E109">A115</font></i></a>, <a class="txt" href="jordan12.html#E14:41_15_2"><i><font color="Green" title="E111">A117</font></i></a>, <a class="txt" href="jordan12.html#E4:41_15_2_9_1_1"><i><font color="Green" title="E123">A137</font></i></a>, <a class="txt" href="jordan12.html#E5:41_15_2_9_1_1"><i><font color="Green" title="E124">A138</font></i></a>, <a class="txt" href="jordan12.html#E6:41_15_2_9_1_1"><i><font color="Green" title="E125">A139</font></i></a>, <a class="ref" href="jordan12.html#T32" target="_self" title="JORDAN12:th.32">Th32</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_2_9_1_2"><b>suppose </b></a><a NAME="E1:41_15_2_9_1_2"/><i><font color="Green" title="E120">A140</font></i>: 
<font color="Maroon" title="c10">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><div class="add"><a NAME="E2:41_15_2_9_1_2"/><i><font color="Green" title="E121">A141</font></i>: 
<font color="Maroon" title="c17">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_15_2_9_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_2_9_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_15_2_9_1_2_1_1"/>
( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">j1</font> or 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j1</font> )
 </span><b>by </b><i><a class="txt" href="jordan12.html#E40:41"><i><font color="Green" title="E69">A32</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_2_9_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:41_15_2_9_1_2_1_1_1"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">j1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_15_2_9_1_2_1_1_1"/>
<font color="Maroon" title="c17">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E22:41_15_2"><i><font color="Green" title="E119">A132</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_2_9_1_2"><i><font color="Green" title="E120">A140</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_2_9_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:41_15_2_9_1_2_1_1_2"/>
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_15_2_9_1_2_1_1_2"/>
<font color="Maroon" title="c17">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E21:41_15_2"><i><font color="Green" title="E118">A130</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_2_9_1_2"><i><font color="Green" title="E120">A140</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:41_15_2_9_1_2"/><i><font color="Green" title="E122">A142</font></i>: 
<font color="Maroon" title="c14">rl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_15_2_9_1_2_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_2_9_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_15_2_9_1_2_2_1"/>
( <font color="Maroon" title="c9">j1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="c2">f</font></span>)</span> or <font color="Maroon" title="c9">j1</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#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span> )
 </span><b>by </b><i><a class="txt" href="jordan12.html#E41:41"><i><font color="Green" title="E70">A33</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_2_9_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:41_15_2_9_1_2_2_1_1"/>
<font color="Maroon" title="c9">j1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="c2">f</font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_15_2_9_1_2_2_1_1"/>
<font color="Maroon" title="c14">rl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E19:41_15_2"><i><font color="Green" title="E116">A126</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_2_9_1_2"><i><font color="Green" title="E120">A140</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_15_2_9_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:41_15_2_9_1_2_2_1_2"/>
<font color="Maroon" title="c9">j1</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#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_15_2_9_1_2_2_1_2"/>
<font color="Maroon" title="c14">rl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E20:41_15_2"><i><font color="Green" title="E117">A128</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_2_9_1_2"><i><font color="Green" title="E120">A140</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E4:41_15_2_9_1_2"/><b>then </b><i><font color="Green" title="E123">A143</font></i>: 
<font color="Maroon" title="c14">rl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E15:41"><i><font color="Green" title="E51">A13</font></i></a></i>;<br/><a NAME="E5:41_15_2_9_1_2"/><i><font color="Green" title="E124">A144</font></i>: 
<font color="Maroon" title="c17">rp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E14:41"><i><font color="Green" title="E50">A11</font></i></a>, <a class="txt" href="jordan12.html#E2:41_15_2_9_1_2"><i><font color="Green" title="E121">A141</font></i></a></i>;<br/><a NAME="E6:41_15_2_9_1_2"/><i><font color="Green" title="E125">A145</font></i>: 
not <font color="Maroon" title="c14">rl</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="c2">f</font>
 <b>by </b><i><a class="txt" href="jordan12.html#E15:41"><i><font color="Green" title="E51">A13</font></i></a>, <a class="txt" href="jordan12.html#E3:41_15_2_9_1_2"><i><font color="Green" title="E122">A142</font></i></a></i>;<br/><a NAME="E7:41_15_2_9_1_2"/>
not <font color="Maroon" title="c17">rp</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="c2">f</font>
 <b>by </b><i><a class="txt" href="jordan12.html#E14:41"><i><font color="Green" title="E50">A11</font></i></a>, <a class="txt" href="jordan12.html#E2:41_15_2_9_1_2"><i><font color="Green" title="E121">A141</font></i></a></i>;<br/><b>hence </b><a NAME="E8:41_15_2_9_1_2"/>
for <font color="Olive" title="b1">C</font> being  <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>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c3">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c4">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 <b>by </b><i><a class="txt" href="jordan12.html#E1:41"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="jordan12.html#E3:41"><i><font color="Green" title="E42">A3</font></i></a>, <a class="txt" href="jordan12.html#E11:41"><i><font color="Green" title="E48">A9</font></i></a>, <a class="txt" href="jordan12.html#E13:41"><i><font color="Green" title="E49">A10</font></i></a>, <a class="txt" href="jordan12.html#E1:41_15_2"><i><font color="Green" title="E106">A112</font></i></a>, <a class="txt" href="jordan12.html#E2:41_15_2"><i><font color="Green" title="E107">A113</font></i></a>, <a class="txt" href="jordan12.html#E8:41_15_2"><i><font color="Green" title="E109">A115</font></i></a>, <a class="txt" href="jordan12.html#E14:41_15_2"><i><font color="Green" title="E111">A117</font></i></a>, <a class="txt" href="jordan12.html#E4:41_15_2_9_1_2"><i><font color="Green" title="E123">A143</font></i></a>, <a class="txt" href="jordan12.html#E5:41_15_2_9_1_2"><i><font color="Green" title="E124">A144</font></i></a>, <a class="txt" href="jordan12.html#E6:41_15_2_9_1_2"><i><font color="Green" title="E125">A145</font></i></a>, <a class="ref" href="jordan12.html#T32" target="_self" title="JORDAN12:th.32">Th32</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E24:41_15_2"/>
for <font color="Olive" title="b1">C</font> being  <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>  holds <br/>(  not <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c3">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c4">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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