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

<span class="kw">let </span><font color="Maroon" title="c1">i</font>, <font color="Maroon" title="c2">j</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">C</font> being   <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>   non  <a href="sppol_1.html#V1" title="SPPOL_1:attr.1">horizontal</a>   non  <a href="sppol_1.html#V2" title="SPPOL_1:attr.2">vertical</a>  <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>  st <font color="Olive" title="b1">C</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  &amp; <font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">j</font> holds <br/> <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Olive" title="b1">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Olive" title="b1">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">C</font> be   <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>   non  <a href="sppol_1.html#V1" title="SPPOL_1:attr.1">horizontal</a>   non  <a href="sppol_1.html#V2" title="SPPOL_1:attr.2">vertical</a>  <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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">C</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  &amp; <font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">j</font> implies  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span></span>)</span> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:59"/><span class="lab"><font color="Green" title="E42">A1</font></span>: 
<font color="Maroon" title="c3">C</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> 
 <span class="kw">and </span><br/><a NAME="E2:59"/><span class="lab"><font color="Green" title="E43">A2</font></span>: 
<font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">j</font>
 <span class="kw">and </span><br/><a NAME="E3:59"/><span class="lab"><font color="Green" title="E44">A3</font></span>: 
 not  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E4:59"/><span class="lab"><font color="Green" title="E45">A4</font></span>: 
 <a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>) <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="ref" href="jordan9.html#D1" title="JORDAN9:def.1">JORDAN9:def 1</a></span>;<br/>
<a NAME="E5:59"/><span class="lab"><font color="Green" title="E46">A5</font></span>: 
 <a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="ref" href="jordan1h.html#T44" target="_self" title="JORDAN1H:th.44">Th52</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E7:59"/><span class="lab"><font color="Green" title="E47">A6</font></span>: 
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E8:59"/><span class="lab"><font color="Green" title="E48">A7</font></span>: 
 not <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59"><span class="lab"><font color="Green" title="E44">A3</font></span></a>, <a class="ref" href="subset_1.html#T2" title="SUBSET_1:th.2">SUBSET_1:2</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">i1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E10:59"/><span class="lab"><font color="Green" title="E49">A8</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i1</font>
 <span class="kw">and </span><br/><a NAME="E11:59"/><span class="lab"><font color="Green" title="E50">A9</font></span>: 
<font color="Maroon" title="c5">i1</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> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E12:59"/><span class="lab"><font color="Green" title="E51">A10</font></span>: 
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span>,<font color="Maroon" title="c5">i1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:59"><span class="lab"><font color="Green" title="E47">A6</font></span></a>, <a class="ref" href="sppol_2.html#T13" title="SPPOL_2:th.13">SPPOL_2:13</a></span>;<br/>
<a NAME="E13:59"/><span class="lab"><font color="Green" title="E52">A11</font></span>: 
 <a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="ref" href="jordan1h.html#T44" target="_self" title="JORDAN1H:th.44">Th52</a></span>;<br/>
<a NAME="E14:59"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A12</font></span>: 
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span>,<font color="Maroon" title="c5">i1</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> (<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span>,<font color="Maroon" title="c5">i1</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:59"><span class="lab"><font color="Green" title="E49">A8</font></span></a>, <a class="txt" href="#E11:59"><span class="lab"><font color="Green" title="E50">A9</font></span></a>, <a class="ref" href="jordan1h.html#T23" target="_self" title="JORDAN1H:th.23">Th29</a></span>;<br/>
<a NAME="E15:59"/><span class="lab"><font color="Green" title="E54">A13</font></span>: 
<font color="Maroon" title="c5">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="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:59"><span class="lab"><font color="Green" title="E50">A9</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:59_1"/>
 ex <font color="Olive" title="b1">i2</font>, <font color="Olive" title="b2">j2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i2</font> &amp; <font color="Olive" title="b1">i2</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> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j2</font> &amp; <font color="Olive" title="b2">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="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span> &amp;  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> (<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span>,<font color="Maroon" title="c5">i1</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span>,<font color="Olive" title="b1">i2</font>,<font color="Olive" title="b2">j2</font>) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c6">f</font> =  <a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>);<br/>
<a NAME="E1:59_1_1"/><span class="lab"><font color="Green" title="E55">A14</font></span>: 
<font color="Maroon" title="c5">i1</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> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:59"><span class="lab"><font color="Green" title="E49">A8</font></span></a>, <a class="txt" href="#E15:59"><span class="lab"><font color="Green" title="E54">A13</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">i4</font>, <font color="Maroon" title="c8">j4</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:59_1_1"/><span class="lab"><font color="Green" title="E56">A15</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">i4</font>,<font color="Maroon" title="c8">j4</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="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:59_1_1"/><span class="lab"><font color="Green" title="E57">A16</font></span>: 
<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">i4</font>,<font color="Maroon" title="c8">j4</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59"><span class="lab"><font color="Green" title="E45">A4</font></span></a>, <a class="ref" href="goboard1.html#D9" title="GOBOARD1:def.9">GOBOARD1:def 9</a></span>;<br/>
<a NAME="E5:59_1_1"/><span class="lab"><font color="Green" title="E58">A17</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i4</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_1_1"><span class="lab"><font color="Green" title="E56">A15</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E6:59_1_1"/><span class="lab"><font color="Green" title="E59">A18</font></span>: 
<font color="Maroon" title="c8">j4</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="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_1_1"><span class="lab"><font color="Green" title="E56">A15</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E7:59_1_1"/><span class="lab"><font color="Green" title="E60">A19</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">j4</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_1_1"><span class="lab"><font color="Green" title="E56">A15</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E8:59_1_1"/><span class="lab"><font color="Green" title="E61">A20</font></span>: 
<font color="Maroon" title="c7">i4</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="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_1_1"><span class="lab"><font color="Green" title="E56">A15</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E9:59_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<a NAME="E10:59_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A21</font></span>: 
<font color="Maroon" title="c5">i1</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> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:59"><span class="lab"><font color="Green" title="E50">A9</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">i5</font>, <font color="Maroon" title="c10">j5</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E12:59_1_1"/><span class="lab"><font color="Green" title="E63">A22</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">i5</font>,<font color="Maroon" title="c10">j5</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="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E13:59_1_1"/><span class="lab"><font color="Green" title="E64">A23</font></span>: 
<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">i1</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="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i5</font>,<font color="Maroon" title="c10">j5</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59"><span class="lab"><font color="Green" title="E45">A4</font></span></a>, <a class="ref" href="goboard1.html#D9" title="GOBOARD1:def.9">GOBOARD1:def 9</a></span>;<br/>
<a NAME="E14:59_1_1"/><span class="lab"><font color="Green" title="E65">A24</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i5</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:59_1_1"><span class="lab"><font color="Green" title="E63">A22</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E15:59_1_1"/>
 <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> (<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span>,<font color="Maroon" title="c5">i1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> (<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span>,<font color="Maroon" title="c5">i1</font>)
 
;<br/>
<a NAME="E16:59_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A25</font></span>: 
( ( <font color="Maroon" title="c7">i4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i5</font> &amp; <font color="Maroon" title="c8">j4</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">j5</font> &amp;  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> (<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span>,<font color="Maroon" title="c5">i1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c7">i4</font>,<font color="Maroon" title="c8">j4</font>) ) or ( <font color="Maroon" title="c7">i4</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="c9">i5</font> &amp; <font color="Maroon" title="c8">j4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j5</font> &amp;  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> (<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span>,<font color="Maroon" title="c5">i1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c7">i4</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">j4</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) ) or ( <font color="Maroon" title="c7">i4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i5</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c8">j4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j5</font> &amp;  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> (<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span>,<font color="Maroon" title="c5">i1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c9">i5</font>,<font color="Maroon" title="c10">j5</font>) ) or ( <font color="Maroon" title="c7">i4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i5</font> &amp; <font color="Maroon" title="c8">j4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j5</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp;  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> (<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span>,<font color="Maroon" title="c5">i1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">i4</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c10">j5</font>) ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:59"><span class="lab"><font color="Green" title="E49">A8</font></span></a>, <a class="txt" href="#E11:59"><span class="lab"><font color="Green" title="E50">A9</font></span></a>, <a class="txt" href="#E13:59"><span class="lab"><font color="Green" title="E52">A11</font></span></a>, <a class="txt" href="#E3:59_1_1"><span class="lab"><font color="Green" title="E56">A15</font></span></a>, <a class="txt" href="#E4:59_1_1"><span class="lab"><font color="Green" title="E57">A16</font></span></a>, <a class="txt" href="#E12:59_1_1"><span class="lab"><font color="Green" title="E63">A22</font></span></a>, <a class="txt" href="#E13:59_1_1"><span class="lab"><font color="Green" title="E64">A23</font></span></a>, <a class="ref" href="goboard5.html#D6" title="GOBOARD5:def.6">GOBOARD5:def 6</a></span>;<br/>
<a NAME="E17:59_1_1"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">i4</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c9">i5</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">j4</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c10">j5</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59"><span class="lab"><font color="Green" title="E45">A4</font></span></a>, <a class="txt" href="#E1:59_1_1"><span class="lab"><font color="Green" title="E55">A14</font></span></a>, <a class="txt" href="#E3:59_1_1"><span class="lab"><font color="Green" title="E56">A15</font></span></a>, <a class="txt" href="#E4:59_1_1"><span class="lab"><font color="Green" title="E57">A16</font></span></a>, <a class="txt" href="#E10:59_1_1"><span class="lab"><font color="Green" title="E62">A21</font></span></a>, <a class="txt" href="#E12:59_1_1"><span class="lab"><font color="Green" title="E63">A22</font></span></a>, <a class="txt" href="#E13:59_1_1"><span class="lab"><font color="Green" title="E64">A23</font></span></a>, <a class="ref" href="goboard1.html#D9" title="GOBOARD1:def.9">GOBOARD1:def 9</a></span>;<br/>
<a NAME="E18:59_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A26</font></span>: 
( (  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i4</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c9">i5</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c8">j4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j5</font> ) or (  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">j4</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c10">j5</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c7">i4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i5</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seqm_3.html#T42" title="SEQM_3:th.42">SEQM_3:42</a></span>;<br/>
<a NAME="E19:59_1_1"/><span class="lab"><font color="Green" title="E68">A27</font></span>: 
<font color="Maroon" title="c10">j5</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="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:59_1_1"><span class="lab"><font color="Green" title="E63">A22</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E20:59_1_1"/><span class="lab"><font color="Green" title="E69">A28</font></span>: 
<font color="Maroon" title="c9">i5</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="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:59_1_1"><span class="lab"><font color="Green" title="E63">A22</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E21:59_1_1"/><span class="lab"><font color="Green" title="E70">A29</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">j5</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:59_1_1"><span class="lab"><font color="Green" title="E63">A22</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:59_1_1_1"/>
( ( <font color="Maroon" title="c7">i4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i5</font> &amp; <font color="Maroon" title="c8">j4</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">j5</font> ) or ( <font color="Maroon" title="c7">i4</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="c9">i5</font> &amp; <font color="Maroon" title="c8">j4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j5</font> ) or ( <font color="Maroon" title="c7">i4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i5</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c8">j4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j5</font> ) or ( <font color="Maroon" title="c7">i4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i5</font> &amp; <font color="Maroon" title="c8">j4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j5</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:59_1_1"><span class="lab"><font color="Green" title="E67">A26</font></span></a>, <a class="ref" href="seqm_3.html#T41" title="SEQM_3:th.41">SEQM_3:41</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:59_1_1_1_1"/><span class="lab"><font color="Green" title="E71">A30</font></span>: 
( <font color="Maroon" title="c7">i4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i5</font> &amp; <font color="Maroon" title="c8">j4</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">j5</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i2</font>, <font color="Olive" title="b2">j2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i2</font> &amp; <font color="Olive" title="b1">i2</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> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j2</font> &amp; <font color="Olive" title="b2">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="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span> &amp;  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> (<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span>,<font color="Maroon" title="c5">i1</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span>,<font color="Olive" title="b1">i2</font>,<font color="Olive" title="b2">j2</font>) )</span><br/><div class="add"><a NAME="E2:59_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">i4</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="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="txt" href="#E10:59"><span class="lab"><font color="Green" title="E49">A8</font></span></a>, <a class="txt" href="#E11:59"><span class="lab"><font color="Green" title="E50">A9</font></span></a>, <a class="txt" href="#E3:59_1_1"><span class="lab"><font color="Green" title="E56">A15</font></span></a>, <a class="txt" href="#E4:59_1_1"><span class="lab"><font color="Green" title="E57">A16</font></span></a>, <a class="txt" href="#E12:59_1_1"><span class="lab"><font color="Green" title="E63">A22</font></span></a>, <a class="txt" href="#E13:59_1_1"><span class="lab"><font color="Green" title="E64">A23</font></span></a>, <a class="ref" href="jordan10.html#T1" title="JORDAN10:th.1">JORDAN10:1</a></span>;<br/><a NAME="E3:59_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">i4</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> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E4:59_1_1_1_1"/>
 ex <font color="Olive" title="b1">i2</font>, <font color="Olive" title="b2">j2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i2</font> &amp; <font color="Olive" title="b1">i2</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> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j2</font> &amp; <font color="Olive" title="b2">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="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span> &amp;  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> (<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span>,<font color="Maroon" title="c5">i1</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span>,<font color="Olive" title="b1">i2</font>,<font color="Olive" title="b2">j2</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="txt" href="#E13:59"><span class="lab"><font color="Green" title="E52">A11</font></span></a>, <a class="txt" href="#E5:59_1_1"><span class="lab"><font color="Green" title="E58">A17</font></span></a>, <a class="txt" href="#E7:59_1_1"><span class="lab"><font color="Green" title="E60">A19</font></span></a>, <a class="txt" href="#E19:59_1_1"><span class="lab"><font color="Green" title="E68">A27</font></span></a>, <a class="txt" href="#E16:59_1_1"><span class="lab"><font color="Green" title="E66">A25</font></span></a>, <a class="txt" href="#E1:59_1_1_1_1"><span class="lab"><font color="Green" title="E71">A30</font></span></a>, <a class="ref" href="jordan1h.html#T38" target="_self" title="JORDAN1H:th.38">Th44</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:59_1_1_1_2"/><span class="lab"><font color="Green" title="E71">A31</font></span>: 
( <font color="Maroon" title="c7">i4</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="c9">i5</font> &amp; <font color="Maroon" title="c8">j4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j5</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i2</font>, <font color="Olive" title="b2">j2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i2</font> &amp; <font color="Olive" title="b1">i2</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> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j2</font> &amp; <font color="Olive" title="b2">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="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span> &amp;  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> (<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span>,<font color="Maroon" title="c5">i1</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span>,<font color="Olive" title="b1">i2</font>,<font color="Olive" title="b2">j2</font>) )</span><br/><div class="add"><a NAME="E2:59_1_1_1_2"/><span class="kw">then </span>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">j4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="txt" href="#E10:59"><span class="lab"><font color="Green" title="E49">A8</font></span></a>, <a class="txt" href="#E11:59"><span class="lab"><font color="Green" title="E50">A9</font></span></a>, <a class="txt" href="#E3:59_1_1"><span class="lab"><font color="Green" title="E56">A15</font></span></a>, <a class="txt" href="#E4:59_1_1"><span class="lab"><font color="Green" title="E57">A16</font></span></a>, <a class="txt" href="#E12:59_1_1"><span class="lab"><font color="Green" title="E63">A22</font></span></a>, <a class="txt" href="#E13:59_1_1"><span class="lab"><font color="Green" title="E64">A23</font></span></a>, <a class="ref" href="jordan10.html#T3" title="JORDAN10:th.3">JORDAN10:3</a></span>;<br/><a NAME="E3:59_1_1_1_2"/><span class="kw">then </span>
1 <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> <font color="Maroon" title="c8">j4</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E4:59_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A32</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">j4</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan5b.html#T2" title="JORDAN5B:th.2">JORDAN5B:2</a></span>;<br/><a NAME="E5:59_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">j4</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <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="c8">j4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:59_1_1"><span class="lab"><font color="Green" title="E60">A19</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>;<br/><span class="kw">hence </span><a NAME="E6:59_1_1_1_2"/>
 ex <font color="Olive" title="b1">i2</font>, <font color="Olive" title="b2">j2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i2</font> &amp; <font color="Olive" title="b1">i2</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> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j2</font> &amp; <font color="Olive" title="b2">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="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span> &amp;  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> (<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span>,<font color="Maroon" title="c5">i1</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span>,<font color="Olive" title="b1">i2</font>,<font color="Olive" title="b2">j2</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="txt" href="#E13:59"><span class="lab"><font color="Green" title="E52">A11</font></span></a>, <a class="txt" href="#E5:59_1_1"><span class="lab"><font color="Green" title="E58">A17</font></span></a>, <a class="txt" href="#E6:59_1_1"><span class="lab"><font color="Green" title="E59">A18</font></span></a>, <a class="txt" href="#E20:59_1_1"><span class="lab"><font color="Green" title="E69">A28</font></span></a>, <a class="txt" href="#E16:59_1_1"><span class="lab"><font color="Green" title="E66">A25</font></span></a>, <a class="txt" href="#E1:59_1_1_1_2"><span class="lab"><font color="Green" title="E71">A31</font></span></a>, <a class="txt" href="#E4:59_1_1_1_2"><span class="lab"><font color="Green" title="E72">A32</font></span></a>, <a class="ref" href="jordan1h.html#T38" target="_self" title="JORDAN1H:th.38">Th44</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:59_1_1_1_3"/><span class="lab"><font color="Green" title="E71">A33</font></span>: 
( <font color="Maroon" title="c7">i4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i5</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c8">j4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j5</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i2</font>, <font color="Olive" title="b2">j2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i2</font> &amp; <font color="Olive" title="b1">i2</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> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j2</font> &amp; <font color="Olive" title="b2">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="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span> &amp;  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> (<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span>,<font color="Maroon" title="c5">i1</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span>,<font color="Olive" title="b1">i2</font>,<font color="Olive" title="b2">j2</font>) )</span><br/><div class="add"><a NAME="E2:59_1_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c10">j5</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="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="txt" href="#E10:59"><span class="lab"><font color="Green" title="E49">A8</font></span></a>, <a class="txt" href="#E11:59"><span class="lab"><font color="Green" title="E50">A9</font></span></a>, <a class="txt" href="#E3:59_1_1"><span class="lab"><font color="Green" title="E56">A15</font></span></a>, <a class="txt" href="#E4:59_1_1"><span class="lab"><font color="Green" title="E57">A16</font></span></a>, <a class="txt" href="#E12:59_1_1"><span class="lab"><font color="Green" title="E63">A22</font></span></a>, <a class="txt" href="#E13:59_1_1"><span class="lab"><font color="Green" title="E64">A23</font></span></a>, <a class="ref" href="jordan10.html#T4" title="JORDAN10:th.4">JORDAN10:4</a></span>;<br/><a NAME="E3:59_1_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c10">j5</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="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E4:59_1_1_1_3"/>
 ex <font color="Olive" title="b1">i2</font>, <font color="Olive" title="b2">j2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i2</font> &amp; <font color="Olive" title="b1">i2</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> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j2</font> &amp; <font color="Olive" title="b2">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="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span> &amp;  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> (<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span>,<font color="Maroon" title="c5">i1</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span>,<font color="Olive" title="b1">i2</font>,<font color="Olive" title="b2">j2</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="txt" href="#E13:59"><span class="lab"><font color="Green" title="E52">A11</font></span></a>, <a class="txt" href="#E8:59_1_1"><span class="lab"><font color="Green" title="E61">A20</font></span></a>, <a class="txt" href="#E14:59_1_1"><span class="lab"><font color="Green" title="E65">A24</font></span></a>, <a class="txt" href="#E21:59_1_1"><span class="lab"><font color="Green" title="E70">A29</font></span></a>, <a class="txt" href="#E16:59_1_1"><span class="lab"><font color="Green" title="E66">A25</font></span></a>, <a class="txt" href="#E1:59_1_1_1_3"><span class="lab"><font color="Green" title="E71">A33</font></span></a>, <a class="ref" href="jordan1h.html#T38" target="_self" title="JORDAN1H:th.38">Th44</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:59_1_1_1_4"/><span class="lab"><font color="Green" title="E71">A34</font></span>: 
( <font color="Maroon" title="c7">i4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i5</font> &amp; <font color="Maroon" title="c8">j4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j5</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i2</font>, <font color="Olive" title="b2">j2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i2</font> &amp; <font color="Olive" title="b1">i2</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> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j2</font> &amp; <font color="Olive" title="b2">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="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span> &amp;  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> (<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span>,<font color="Maroon" title="c5">i1</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span>,<font color="Olive" title="b1">i2</font>,<font color="Olive" title="b2">j2</font>) )</span><br/><div class="add"><a NAME="E2:59_1_1_1_4"/><span class="kw">then </span>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">i4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="txt" href="#E10:59"><span class="lab"><font color="Green" title="E49">A8</font></span></a>, <a class="txt" href="#E11:59"><span class="lab"><font color="Green" title="E50">A9</font></span></a>, <a class="txt" href="#E3:59_1_1"><span class="lab"><font color="Green" title="E56">A15</font></span></a>, <a class="txt" href="#E4:59_1_1"><span class="lab"><font color="Green" title="E57">A16</font></span></a>, <a class="txt" href="#E12:59_1_1"><span class="lab"><font color="Green" title="E63">A22</font></span></a>, <a class="txt" href="#E13:59_1_1"><span class="lab"><font color="Green" title="E64">A23</font></span></a>, <a class="ref" href="jordan10.html#T2" title="JORDAN10:th.2">JORDAN10:2</a></span>;<br/><a NAME="E3:59_1_1_1_4"/><span class="kw">then </span>
1 <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> <font color="Maroon" title="c7">i4</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E4:59_1_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A35</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i4</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan5b.html#T2" title="JORDAN5B:th.2">JORDAN5B:2</a></span>;<br/><a NAME="E5:59_1_1_1_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">i4</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <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="c7">i4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:59_1_1"><span class="lab"><font color="Green" title="E58">A17</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>;<br/><span class="kw">hence </span><a NAME="E6:59_1_1_1_4"/>
 ex <font color="Olive" title="b1">i2</font>, <font color="Olive" title="b2">j2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i2</font> &amp; <font color="Olive" title="b1">i2</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> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j2</font> &amp; <font color="Olive" title="b2">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="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span> &amp;  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> (<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span>,<font color="Maroon" title="c5">i1</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span>,<font color="Olive" title="b1">i2</font>,<font color="Olive" title="b2">j2</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="txt" href="#E13:59"><span class="lab"><font color="Green" title="E52">A11</font></span></a>, <a class="txt" href="#E8:59_1_1"><span class="lab"><font color="Green" title="E61">A20</font></span></a>, <a class="txt" href="#E6:59_1_1"><span class="lab"><font color="Green" title="E59">A18</font></span></a>, <a class="txt" href="#E21:59_1_1"><span class="lab"><font color="Green" title="E70">A29</font></span></a>, <a class="txt" href="#E16:59_1_1"><span class="lab"><font color="Green" title="E66">A25</font></span></a>, <a class="txt" href="#E1:59_1_1_1_4"><span class="lab"><font color="Green" title="E71">A34</font></span></a>, <a class="txt" href="#E4:59_1_1_1_4"><span class="lab"><font color="Green" title="E72">A35</font></span></a>, <a class="ref" href="jordan1h.html#T38" target="_self" title="JORDAN1H:th.38">Th44</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">i2</font>, <font color="Maroon" title="c7">j2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:59_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">i2</font>
 <span class="kw">and </span><br/><a NAME="E4:59_1"/><span class="lab"><font color="Green" title="E55">A36</font></span>: 
<font color="Maroon" title="c6">i2</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> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:59_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">j2</font>
 <span class="kw">and </span><br/><a NAME="E6:59_1"/><span class="lab"><font color="Green" title="E56">A37</font></span>: 
<font color="Maroon" title="c7">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="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E7:59_1"/><span class="lab"><font color="Green" title="E57">A38</font></span>: 
 <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> (<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span>,<font color="Maroon" title="c5">i1</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span>,<font color="Maroon" title="c6">i2</font>,<font color="Maroon" title="c7">j2</font>)
 ;<br/><a NAME="E8:59_1"/><span class="lab"><font color="Green" title="E58">A39</font></span>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> (<span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span>,<font color="Maroon" title="c5">i1</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span>,<font color="Maroon" title="c6">i2</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:59_1"><span class="lab"><font color="Green" title="E57">A38</font></span></a>, <a class="ref" href="tops_1.html#T19" title="TOPS_1:th.19">TOPS_1:19</a></span>;<br/><a NAME="E9:59_1"/><span class="lab"><font color="Green" title="E59">A40</font></span>: 
 <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="goboard9.html#D2" title="GOBOARD9:def.2">GOBOARD9:def 2</a></span>;<br/><span class="lab"><font color="Green" title="E60">A41</font></span>: <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p2">(<span class="default"><a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="gobrd14.html#T22" title="GOBRD14:th.22">GOBRD14:22</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></span>
<br/>.= 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="gobrd14.html#T15" title="GOBRD14:th.15">GOBRD14:15</a></span>
;<br/><span class="kw">assume </span><a NAME="E11:59_1"/>
 not  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> (<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span>,<font color="Maroon" title="c5">i1</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E12:59_1"/><span class="kw">then </span>
 not  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span>,<font color="Maroon" title="c6">i2</font>,<font color="Maroon" title="c7">j2</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:59_1"><span class="lab"><font color="Green" title="E57">A38</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E13:59_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A42</font></span>: 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span>,<font color="Maroon" title="c6">i2</font>,<font color="Maroon" title="c7">j2</font>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:59_1"><span class="lab"><font color="Green" title="E60">A41</font></span></a>, <a class="ref" href="xboole_1.html#T73" title="XBOOLE_1:th.73">XBOOLE_1:73</a></span>;<br/><a NAME="E14:59_1"/><span class="lab"><font color="Green" title="E62">A43</font></span>: 
( <font color="Maroon" title="c6">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="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span> &amp; <font color="Maroon" title="c7">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="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59_1"><span class="lab"><font color="Green" title="E55">A36</font></span></a>, <a class="txt" href="#E6:59_1"><span class="lab"><font color="Green" title="E56">A37</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E15:59_1"/><span class="kw">then </span>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span>,<font color="Maroon" title="c6">i2</font>,<font color="Maroon" title="c7">j2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p2">(<span class="default"><a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span>,<font color="Maroon" title="c6">i2</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gobrd11.html#T35" title="GOBRD11:th.35">GOBRD11:35</a></span>;<br/><a NAME="E16:59_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A44</font></span>: 
 <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="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span>,<font color="Maroon" title="c6">i2</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:59_1"><span class="lab"><font color="Green" title="E61">A42</font></span></a>, <a class="ref" href="tsep_1.html#T36" title="TSEP_1:th.36">TSEP_1:36</a></span>;<br/><a NAME="E17:59_1"/>
 <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="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span>,<font color="Maroon" title="c6">i2</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:59"><span class="lab"><font color="Green" title="E46">A5</font></span></a>, <a class="txt" href="#E14:59_1"><span class="lab"><font color="Green" title="E62">A43</font></span></a>, <a class="ref" href="gobrd12.html#T1" title="GOBRD12:th.1">GOBRD12:1</a></span>;<br/><a NAME="E18:59_1"/><span class="kw">then </span>
 <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="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span>,<font color="Maroon" title="c6">i2</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:59_1"><span class="lab"><font color="Green" title="E62">A43</font></span></a>, <a class="txt" href="#E16:59_1"><span class="lab"><font color="Green" title="E63">A44</font></span></a>, <a class="txt" href="#E9:59_1"><span class="lab"><font color="Green" title="E59">A40</font></span></a>, <a class="ref" href="goboard9.html#T4" title="GOBOARD9:th.4">GOBOARD9:4</a>, <a class="ref" href="goboard9.html#T18" title="GOBOARD9:th.18">GOBOARD9:18</a></span>;<br/><a NAME="E19:59_1"/><span class="kw">then </span>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> (<span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span>,<font color="Maroon" title="c5">i1</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:59_1"><span class="lab"><font color="Green" title="E58">A39</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E20:59_1"/><span class="kw">then </span>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p2">(<span class="default"><a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> (<span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span>,<font color="Maroon" title="c5">i1</font>)</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T19" title="PRE_TOPC:th.19">PRE_TOPC:19</a></span>;<br/><a NAME="E21:59_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A45</font></span>: 
 <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> (<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span>,<font color="Maroon" title="c5">i1</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59"><span class="lab"><font color="Green" title="E45">A4</font></span></a>, <a class="txt" href="#E10:59"><span class="lab"><font color="Green" title="E49">A8</font></span></a>, <a class="txt" href="#E11:59"><span class="lab"><font color="Green" title="E50">A9</font></span></a>, <a class="txt" href="#E14:59"><span class="lab"><font color="Green" title="E53">A12</font></span></a>, <a class="ref" href="jordan9.html#T11" title="JORDAN9:th.11">JORDAN9:11</a></span>;<br/><a NAME="E22:59_1"/>
(  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span>,<font color="Maroon" title="c5">i1</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span>,<font color="Maroon" title="c5">i1</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span>) &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span>,<font color="Maroon" title="c5">i1</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> (<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span>,<font color="Maroon" title="c5">i1</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59"><span class="lab"><font color="Green" title="E45">A4</font></span></a>, <a class="txt" href="#E10:59"><span class="lab"><font color="Green" title="E49">A8</font></span></a>, <a class="txt" href="#E11:59"><span class="lab"><font color="Green" title="E50">A9</font></span></a>, <a class="ref" href="jordan1h.html#T22" target="_self" title="JORDAN1H:th.22">Th28</a>, <a class="ref" href="gobrd13.html#T33" title="GOBRD13:th.33">GOBRD13:33</a></span>;<br/><a NAME="E23:59_1"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span>,<font color="Maroon" title="c5">i1</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> (<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span>,<font color="Maroon" title="c5">i1</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E24:59_1"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">j</font>)</span>)</span>,<font color="Maroon" title="c5">i1</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:59_1"><span class="lab"><font color="Green" title="E64">A45</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><span class="kw">hence </span><a NAME="E25:59_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:59"><span class="lab"><font color="Green" title="E48">A7</font></span></a>, <a class="txt" href="#E12:59"><span class="lab"><font color="Green" title="E51">A10</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E17:59"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A46</font></span>: 
<font color="Maroon" title="c3">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="txt" href="#E10:59"><span class="lab"><font color="Green" title="E49">A8</font></span></a>, <a class="txt" href="#E11:59"><span class="lab"><font color="Green" title="E50">A9</font></span></a>, <a class="txt" href="#E14:59"><span class="lab"><font color="Green" title="E53">A12</font></span></a>, <a class="ref" href="jordan9.html#T31" title="JORDAN9:th.31">JORDAN9:31</a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></span>;<br/>
<a NAME="E18:59"/>
(  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span></span>)</span> &amp; <font color="Maroon" title="c3">C</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="ref" href="gobrd14.html#T22" title="GOBRD14:th.22">GOBRD14:22</a>, <a class="ref" href="jordan10.html#T5" title="JORDAN10:th.5">JORDAN10:5</a></span>;<br/>
<a NAME="E19:59"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A47</font></span>: 
<font color="Maroon" title="c3">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:59"><span class="lab"><font color="Green" title="E55">A46</font></span></a>, <a class="ref" href="xboole_1.html#T70" title="XBOOLE_1:th.70">XBOOLE_1:70</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">D</font> = <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  as   <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> ;<br/>
<a NAME="E21:59"/>
<font color="Maroon" title="c6">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gobrd12.html#T10" title="GOBRD12:th.10">GOBRD12:10</a></span>;<br/>
<a NAME="E22:59"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A48</font></span>: 
 <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<a NAME="E23:59"/>
<font color="Maroon" title="c3">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="ref" href="jordan10.html#T11" title="JORDAN10:th.11">JORDAN10:11</a></span>;<br/>
<a NAME="E24:59"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A49</font></span>: 
<font color="Maroon" title="c3">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:59"><span class="lab"><font color="Green" title="E57">A48</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E25:59"/>
<font color="Maroon" title="c3">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a> <font color="Maroon" title="c3">C</font>
 
;<br/>
<a NAME="E26:59"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A50</font></span>: 
<font color="Maroon" title="c3">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="ref" href="jordan10.html#T11" title="JORDAN10:th.11">JORDAN10:11</a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></span>;<br/>
<a NAME="E27:59"/>
(  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <font color="Maroon" title="c6">D</font> &amp;  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">i</font>)</span>)</span> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <font color="Maroon" title="c6">D</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="goboard9.html#D1" title="GOBOARD9:def.1">GOBOARD9:def 1</a>, <a class="ref" href="goboard9.html#D2" title="GOBOARD9:def.2">GOBOARD9:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E28:59"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="txt" href="#E19:59"><span class="lab"><font color="Green" title="E56">A47</font></span></a>, <a class="txt" href="#E24:59"><span class="lab"><font color="Green" title="E58">A49</font></span></a>, <a class="txt" href="#E26:59"><span class="lab"><font color="Green" title="E59">A50</font></span></a>, <a class="ref" href="jordan9.html#T1" title="JORDAN9:th.1">JORDAN9:1</a>, <a class="ref" href="sprect_4.html#T6" title="SPRECT_4:th.6">SPRECT_4:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
