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<span class="kw">let </span><font color="Maroon" title="c1">i</font>, <font color="Maroon" title="c2">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="Maroon" title="c1">i</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="Olive" title="b1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> holds <br/> <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="Olive" title="b1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">i</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Olive" title="b1">C</font></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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">i</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">n</font>)</span>)</span> implies  <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="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">i</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c3">C</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:62"/><span class="lab"><font color="Green" title="E38">A1</font></span>: 
<font color="Maroon" title="c1">i</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">n</font>)</span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">i</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c3">C</font></span><br/>

<a NAME="E2:62"/><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="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">i</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan8.html#T17" onmouseover="rs('jordan8/T17')" onmouseout="rh()">JORDAN8:17</a></span>;<br/>
<a NAME="E3:62"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A2</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="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">i</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T23" onmouseover="rs('subset_1/T23')" onmouseout="rh()">SUBSET_1:23</a></span>;<br/>
<a NAME="E4:62"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_0.html#K1" title="MATRIX_0: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">n</font>)</span>)</span>
 
;<br/>
<a NAME="E5:62"/><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="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">i</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  &amp;  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="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">i</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="ref" href="jordan1a.html#T24" target="_self" onmouseover="rs('jordan1a/T24')" onmouseout="rh()">Th24</a>, <a class="ref" href="jordan1a.html#T25" target="_self" onmouseover="rs('jordan1a/T25')" onmouseout="rh()">Th25</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E7:62"/><span class="lab"><font color="Green" title="E40">A3</font></span>: 
<font color="Maroon" title="c4">W</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <font color="Maroon" title="c3">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">and </span><br/><a NAME="E8:62"/><span class="lab"><font color="Green" title="E41">A4</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="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">i</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:62"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="ref" href="goboard9.html#T3" onmouseover="rs('goboard9/T3')" onmouseout="rh()">GOBOARD9:3</a></span>;<br/>
<a NAME="E9:62"/>
 not <font color="Maroon" title="c4">W</font> is  <a href="rltopsp1.html#V9" title="RLTOPSP1:attr.9">bounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="txt" href="#E8:62"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="ref" href="jordan1a.html#T26" target="_self" onmouseover="rs('jordan1a/T26')" onmouseout="rh()">Th26</a>, <a class="ref" href="rltopsp1.html#T42" onmouseover="rs('rltopsp1/T42')" onmouseout="rh()">RLTOPSP1:42</a></span>;<br/>
<a NAME="E10:62"/><span class="kw">then </span>
<font color="Maroon" title="c4">W</font> <a href="jordan2c.html#R2" title="JORDAN2C:pred.2">is_outside_component_of</a> <font color="Maroon" title="c3">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:62"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="ref" href="jordan2c.html#D3" onmouseover="rs('jordan2c/D3')" onmouseout="rh()">JORDAN2C:def 3</a></span>;<br/>
<a NAME="E11:62"/><span class="kw">then </span>
<font color="Maroon" title="c4">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c3">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan2c.html#T23" onmouseover="rs('jordan2c/T23')" onmouseout="rh()">JORDAN2C:23</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:62"/>
 <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="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">i</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:62"><span class="lab"><font color="Green" title="E41">A4</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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