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<b>let </b><font color="Maroon" title="c1">p</font>, <font color="Maroon" title="c2">p1</font>, <font color="Maroon" title="c3">p2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon" title="c4">f</font> be  non <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> <a href="goboard5.html#V2" title="GOBOARD5:attr.2">standard</a> <a href="goboard5.html#NM1" title="GOBOARD5:NM.1">special_circular_sequence</a>;<br/>







<b>assume </b><a NAME="E1:39"/><i><font color="Green" title="E38">A1</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">p</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>

<b>let </b><font color="Maroon" title="c5">r</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:39"/><i><font color="Green" title="E39">A2</font></i>: 
not <font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <b>and </b><br/><a NAME="E3:39"/><i><font color="Green" title="E40">A3</font></i>: 
( not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font> &amp; not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font> )
 <b>and </b><br/><a NAME="E4:39"/><i><font color="Green" title="E41">A4</font></i>: 
( ( <font color="Maroon" title="c2">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c2">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) or ( <font color="Maroon" title="c2">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c2">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) )
 <b>and </b><br/><a NAME="E5:39"/><i><font color="Green" title="E42">A5</font></i>: 
 ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> &amp; ( <font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c4">f</font>,<font color="Olive" title="b1">i</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c4">f</font></span>)</span> or <font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c4">f</font>,<font color="Olive" title="b1">i</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c4">f</font></span>)</span> ) &amp; <font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Olive" title="b1">i</font> )
 <b>and </b><br/><a NAME="E6:39"/><i><font color="Green" title="E43">A6</font></i>: 
not <font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font>
 ;<br/>

<a NAME="E7:39"/><i><font color="Green" title="E44">A7</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c4">f</font>
 
<b>by </b><i><a class="ref" href="goboard5.html#D5" title="GOBOARD5:def.5">GOBOARD5:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E9:39"/><i><font color="Green" title="E45">A8</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> &amp; ( <font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">i</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c4">f</font></span>)</span> or <font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">i</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c4">f</font></span>)</span> ) &amp; <font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">i</font> )
 <b>by </b><i><a class="txt" href="jordan12.html#E5:39"><i><font color="Green" title="E42">A5</font></i></a></i>;<br/>
<a NAME="E10:39"/><i><font color="Green" title="E46">A9</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">i</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c4">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan12.html#E7:39"><i><font color="Green" title="E44">A7</font></i></a>, <a class="txt" href="jordan12.html#E9:39"><i><font color="Green" title="E45">A8</font></i></a>, <a class="ref" href="jordan1h.html#T26" title="JORDAN1H:th.26">JORDAN1H:26</a></i>;<br/>
<a NAME="E11:39"/><i><font color="Green" title="E47">A10</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">i</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c4">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan12.html#E7:39"><i><font color="Green" title="E44">A7</font></i></a>, <a class="txt" href="jordan12.html#E9:39"><i><font color="Green" title="E45">A8</font></i></a>, <a class="ref" href="jordan1h.html#T28" title="JORDAN1H:th.28">JORDAN1H:28</a></i>;<br/>
<a NAME="E12:39"/><i><font color="Green" title="E48">A11</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan12.html#E1:39"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E13:39"/><b>then </b><i><font color="Green" title="E49">A12</font></i>: 
( <font color="Maroon" title="c1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p2</font> &amp; <font color="Maroon" title="c1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">p1</font> )
 
<b>by </b><i><a class="txt" href="jordan12.html#E3:39"><i><font color="Green" title="E40">A3</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E14:39"/><i><font color="Green" title="E50">A13</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 
<b>by </b><i><a class="txt" href="jordan12.html#E12:39"><i><font color="Green" title="E48">A11</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E15:39"/><i><font color="Green" title="E51">A14</font></i>: 
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">i</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c4">f</font></span>)</span> is <a href="jordan1.html#V1" title="JORDAN1:attr.1">convex</a>
 
<b>by </b><i><a class="txt" href="jordan12.html#E9:39"><i><font color="Green" title="E45">A8</font></i></a>, <a class="ref" href="jordan12.html#T23" target="_self" title="JORDAN12:th.23">Th23</a></i>;<br/>
<a NAME="E16:39"/><i><font color="Green" title="E52">A15</font></i>: 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">i</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c4">f</font></span>)</span> is <a href="jordan1.html#V1" title="JORDAN1:attr.1">convex</a>
 
<b>by </b><i><a class="txt" href="jordan12.html#E9:39"><i><font color="Green" title="E45">A8</font></i></a>, <a class="ref" href="jordan12.html#T23" target="_self" title="JORDAN12:th.23">Th23</a></i>;<br/>
<a NAME="E17:39"/><i><font color="Green" title="E53">A16</font></i>: 
<span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">i</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</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> <font color="Maroon" title="c4">f</font>
 
<b>by </b><i><a class="txt" href="jordan12.html#E7:39"><i><font color="Green" title="E44">A7</font></i></a>, <a class="txt" href="jordan12.html#E9:39"><i><font color="Green" title="E45">A8</font></i></a>, <a class="ref" href="jordan9.html#T29" title="JORDAN9:th.29">JORDAN9:29</a></i>;<br/>
<a NAME="E18:39"/><i><font color="Green" title="E54">A17</font></i>: 
<span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">i</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c4">f</font>
 
<b>by </b><i><a class="txt" href="jordan12.html#E7:39"><i><font color="Green" title="E44">A7</font></i></a>, <a class="txt" href="jordan12.html#E9:39"><i><font color="Green" title="E45">A8</font></i></a>, <a class="ref" href="jordan9.html#T29" title="JORDAN9:th.29">JORDAN9:29</a></i>;<br/>
<div><i><font color="Green" title="E55">A18</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_1"/>
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">i</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c4">f</font></span>)</span>
 ;<br/><a NAME="E2:39_1"/><b>then </b>
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">i</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E6:39"><i><font color="Green" title="E43">A6</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E3:39_1"/>
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="txt" href="jordan12.html#E17:39"><i><font color="Green" title="E53">A16</font></i></a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E56">A19</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_2"/>
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">i</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c4">f</font></span>)</span>
 ;<br/><a NAME="E2:39_2"/><b>then </b>
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">i</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E6:39"><i><font color="Green" title="E43">A6</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E3:39_2"/>
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="txt" href="jordan12.html#E18:39"><i><font color="Green" title="E54">A17</font></i></a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E57">A20</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_3"/><i><font color="Green" title="E58">A21</font></i>: 
( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c5">r</font>,<font color="Maroon" title="c1">p</font> &amp; <font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">i</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c4">f</font></span>)</span> )
 ;<br/><a NAME="E2:39_3"/><b>then </b>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c5">r</font>,<font color="Maroon" title="c1">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">i</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E9:39"><i><font color="Green" title="E45">A8</font></i></a>, <a class="txt" href="jordan12.html#E11:39"><i><font color="Green" title="E47">A10</font></i></a>, <a class="txt" href="jordan12.html#E15:39"><i><font color="Green" title="E51">A14</font></i></a>, <a class="ref" href="jordan1.html#D1" title="JORDAN1:def.1">JORDAN1:def 1</a></i>;<br/><a NAME="E3:39_3"/><b>then </b>
<font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">i</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E3:39"><i><font color="Green" title="E40">A3</font></i></a>, <a class="txt" href="jordan12.html#E1:39_3"><i><font color="Green" title="E58">A21</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E4:39_3"/>
 ex <font color="Olive" title="b1">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 <b>by </b><i><a class="txt" href="jordan12.html#E17:39"><i><font color="Green" title="E53">A16</font></i></a>, <a class="txt" href="jordan12.html#E19:39"><i><font color="Green" title="E55">A18</font></i></a>, <a class="txt" href="jordan12.html#E1:39_3"><i><font color="Green" title="E58">A21</font></i></a>, <a class="ref" href="jordan12.html#T15" target="_self" title="JORDAN12:th.15">Th15</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E58">A22</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_4"/><i><font color="Green" title="E59">A23</font></i>: 
( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c5">r</font>,<font color="Maroon" title="c1">p</font> &amp; <font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">i</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c4">f</font></span>)</span> )
 ;<br/><a NAME="E2:39_4"/><b>then </b>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c5">r</font>,<font color="Maroon" title="c1">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">i</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E9:39"><i><font color="Green" title="E45">A8</font></i></a>, <a class="txt" href="jordan12.html#E10:39"><i><font color="Green" title="E46">A9</font></i></a>, <a class="txt" href="jordan12.html#E16:39"><i><font color="Green" title="E52">A15</font></i></a>, <a class="ref" href="jordan1.html#D1" title="JORDAN1:def.1">JORDAN1:def 1</a></i>;<br/><a NAME="E3:39_4"/><b>then </b>
<font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">i</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E3:39"><i><font color="Green" title="E40">A3</font></i></a>, <a class="txt" href="jordan12.html#E1:39_4"><i><font color="Green" title="E59">A23</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E4:39_4"/>
 ex <font color="Olive" title="b1">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 <b>by </b><i><a class="txt" href="jordan12.html#E18:39"><i><font color="Green" title="E54">A17</font></i></a>, <a class="txt" href="jordan12.html#E20:39"><i><font color="Green" title="E56">A19</font></i></a>, <a class="txt" href="jordan12.html#E1:39_4"><i><font color="Green" title="E59">A23</font></i></a>, <a class="ref" href="jordan12.html#T15" target="_self" title="JORDAN12:th.15">Th15</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E59">A24</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_5"/><i><font color="Green" title="E60">A25</font></i>: 
( <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c5">r</font>,<font color="Maroon" title="c1">p</font> &amp; <font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">i</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c4">f</font></span>)</span> )
 ;<br/><a NAME="E2:39_5"/><b>then </b>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c5">r</font>,<font color="Maroon" title="c1">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">i</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E9:39"><i><font color="Green" title="E45">A8</font></i></a>, <a class="txt" href="jordan12.html#E11:39"><i><font color="Green" title="E47">A10</font></i></a>, <a class="txt" href="jordan12.html#E15:39"><i><font color="Green" title="E51">A14</font></i></a>, <a class="ref" href="jordan1.html#D1" title="JORDAN1:def.1">JORDAN1:def 1</a></i>;<br/><a NAME="E3:39_5"/><b>then </b>
<font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">i</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E3:39"><i><font color="Green" title="E40">A3</font></i></a>, <a class="txt" href="jordan12.html#E1:39_5"><i><font color="Green" title="E60">A25</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E4:39_5"/>
 ex <font color="Olive" title="b1">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 <b>by </b><i><a class="txt" href="jordan12.html#E17:39"><i><font color="Green" title="E53">A16</font></i></a>, <a class="txt" href="jordan12.html#E19:39"><i><font color="Green" title="E55">A18</font></i></a>, <a class="txt" href="jordan12.html#E1:39_5"><i><font color="Green" title="E60">A25</font></i></a>, <a class="ref" href="jordan12.html#T15" target="_self" title="JORDAN12:th.15">Th15</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E60">A26</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_6"/><i><font color="Green" title="E61">A27</font></i>: 
( <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c5">r</font>,<font color="Maroon" title="c1">p</font> &amp; <font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">i</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c4">f</font></span>)</span> )
 ;<br/><a NAME="E2:39_6"/><b>then </b>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c5">r</font>,<font color="Maroon" title="c1">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">i</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E9:39"><i><font color="Green" title="E45">A8</font></i></a>, <a class="txt" href="jordan12.html#E10:39"><i><font color="Green" title="E46">A9</font></i></a>, <a class="txt" href="jordan12.html#E16:39"><i><font color="Green" title="E52">A15</font></i></a>, <a class="ref" href="jordan1.html#D1" title="JORDAN1:def.1">JORDAN1:def 1</a></i>;<br/><a NAME="E3:39_6"/><b>then </b>
<font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">i</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E3:39"><i><font color="Green" title="E40">A3</font></i></a>, <a class="txt" href="jordan12.html#E1:39_6"><i><font color="Green" title="E61">A27</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E4:39_6"/>
 ex <font color="Olive" title="b1">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 <b>by </b><i><a class="txt" href="jordan12.html#E18:39"><i><font color="Green" title="E54">A17</font></i></a>, <a class="txt" href="jordan12.html#E20:39"><i><font color="Green" title="E56">A19</font></i></a>, <a class="txt" href="jordan12.html#E1:39_6"><i><font color="Green" title="E61">A27</font></i></a>, <a class="ref" href="jordan12.html#T15" target="_self" title="JORDAN12:th.15">Th15</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_7"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_7"/>
( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c5">r</font>,<font color="Maroon" title="c1">p</font> or <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c5">r</font>,<font color="Maroon" title="c1">p</font> )
 </span><b>by </b><i><a class="txt" href="jordan12.html#E2:39"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="jordan12.html#E4:39"><i><font color="Green" title="E41">A4</font></i></a>, <a class="txt" href="jordan12.html#E13:39"><i><font color="Green" title="E49">A12</font></i></a>, <a class="txt" href="jordan12.html#E14:39"><i><font color="Green" title="E50">A13</font></i></a>, <a class="ref" href="jordan12.html#T29" target="_self" title="JORDAN12:th.29">Th29</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_7_1"><b>suppose </b></a><a NAME="E1:39_7_1"/><i><font color="Green" title="E61">A28</font></i>: 
<font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c5">r</font>,<font color="Maroon" title="c1">p</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_7_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_7_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_7_1_1_1"/>
( <font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">i</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c4">f</font></span>)</span> or <font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">i</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c4">f</font></span>)</span> )
 </span><b>by </b><i><a class="txt" href="jordan12.html#E9:39"><i><font color="Green" title="E45">A8</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_7_1_1_1_1"><b>suppose </b></a><a NAME="E1:39_7_1_1_1_1"/>
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">i</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c4">f</font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:39_7_1_1_1_1"/>
(  ex <font color="Olive" title="b1">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> ) or  ex <font color="Olive" title="b1">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> ) )
 <b>by </b><i><a class="txt" href="jordan12.html#E21:39"><i><font color="Green" title="E57">A20</font></i></a>, <a class="txt" href="jordan12.html#E1:39_7_1"><i><font color="Green" title="E61">A28</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_7_1_1_1_2"><b>suppose </b></a><a NAME="E1:39_7_1_1_1_2"/>
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">i</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c4">f</font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:39_7_1_1_1_2"/>
(  ex <font color="Olive" title="b1">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> ) or  ex <font color="Olive" title="b1">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> ) )
 <b>by </b><i><a class="txt" href="jordan12.html#E22:39"><i><font color="Green" title="E58">A22</font></i></a>, <a class="txt" href="jordan12.html#E1:39_7_1"><i><font color="Green" title="E61">A28</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:39_7_1"/>
(  ex <font color="Olive" title="b1">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> ) or  ex <font color="Olive" title="b1">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> ) )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_7_2"><b>suppose </b></a><a NAME="E1:39_7_2"/><i><font color="Green" title="E61">A29</font></i>: 
<font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c5">r</font>,<font color="Maroon" title="c1">p</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_7_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_7_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_7_2_1_1"/>
( <font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">i</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c4">f</font></span>)</span> or <font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">i</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c4">f</font></span>)</span> )
 </span><b>by </b><i><a class="txt" href="jordan12.html#E9:39"><i><font color="Green" title="E45">A8</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_7_2_1_1_1"><b>suppose </b></a><a NAME="E1:39_7_2_1_1_1"/>
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">i</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c4">f</font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:39_7_2_1_1_1"/>
(  ex <font color="Olive" title="b1">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> ) or  ex <font color="Olive" title="b1">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> ) )
 <b>by </b><i><a class="txt" href="jordan12.html#E23:39"><i><font color="Green" title="E59">A24</font></i></a>, <a class="txt" href="jordan12.html#E1:39_7_2"><i><font color="Green" title="E61">A29</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_7_2_1_1_2"><b>suppose </b></a><a NAME="E1:39_7_2_1_1_2"/>
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">i</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c4">f</font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:39_7_2_1_1_2"/>
(  ex <font color="Olive" title="b1">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> ) or  ex <font color="Olive" title="b1">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> ) )
 <b>by </b><i><a class="txt" href="jordan12.html#E24:39"><i><font color="Green" title="E60">A26</font></i></a>, <a class="txt" href="jordan12.html#E1:39_7_2"><i><font color="Green" title="E61">A29</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:39_7_2"/>
(  ex <font color="Olive" title="b1">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> ) or  ex <font color="Olive" title="b1">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> ) )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
