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<b>let </b><font color="Maroon">f</font> be  non <a href="funct_1.html#V3">constant</a> <a href="goboard5.html#V2">standard</a> <a href="goboard5.html#NM1">special_circular_sequence</a>;<br/><b>let </b><font color="Maroon">p1</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">p2</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>







<b>assume </b><a NAME="E1:40"/><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<b>let </b><font color="Maroon">rl</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">rp</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:40"/><i><font color="Green">E43</font></i>: 
( not <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font> &amp; not <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font> )
 <b>and </b><br/><a NAME="E3:40"/><i><font color="Green">E45</font></i>: 
( ( <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">rl</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">rl</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">rp</font> <a href="euclid.html#K21">`1</a>  ) or ( <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">rl</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">rl</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">rp</font> <a href="euclid.html#K22">`2</a>  ) )
 <b>and </b><br/><a NAME="E4:40"/><i><font color="Green">E54</font></i>: 
 ex <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> &amp; <font color="Maroon">rl</font> <a href="hidden.html#R2">in</a>  <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Olive">i</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">rp</font> <a href="hidden.html#R2">in</a>  <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Olive">i</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">i</font> )
 <b>and </b><br/><a NAME="E5:40"/><i><font color="Green">E55</font></i>: 
not <font color="Maroon">rl</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E6:40"/><i><font color="Green">E56</font></i>: 
not <font color="Maroon">rp</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E7:40"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">f</font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="goboard5.html#D5">GOBOARD5:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E9:40"/><i><font color="Green">E58</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> &amp; <font color="Maroon">rl</font> <a href="hidden.html#R2">in</a>  <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">i</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">rp</font> <a href="hidden.html#R2">in</a>  <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">i</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan12.html#E1:40"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E10:40"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">rl</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">i</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan12.html#E2:40"><i><font color="Green">E43</font></i></a>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E11:40"/><i><font color="Green">E60</font></i>: 
<span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">i</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="txt" href="jordan12.html#E3:40"><i><font color="Green">E45</font></i></a>, , <a class="ref" href="jordan9.html#T29">JORDAN9:29</a></i>;<br/>
<a NAME="E12:40"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">rp</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">i</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E13:40"/><i><font color="Green">E62</font></i>: 
<span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">i</font>,<span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="txt" href="jordan12.html#E3:40"><i><font color="Green">E45</font></i></a>, , <a class="ref" href="jordan9.html#T29">JORDAN9:29</a></i>;<br/>
<a NAME="E14:40"/><i><font color="Green">E63</font></i>: 
not <font color="Maroon">rl</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="txt" href="jordan12.html#E2"><i><font color="Green">Lemma47</font></i></a>, , <a class="ref" href="gobrd14.html#T27">GOBRD14:27</a></i>;<br/>
<a NAME="E15:40"/><i><font color="Green">E64</font></i>: 
not <font color="Maroon">rp</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="txt" href="jordan12.html#E3"><i><font color="Green">Lemma49</font></i></a>, <a class="ref" href="jordan12.html#T3" target="_self">Th3</a>, <a class="ref" href="gobrd14.html#T28">GOBRD14:28</a></i>;<br/>
<div><i><font color="Green">E65</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:40_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">rl</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_1_1"/>
(  ex <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">rl</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> &amp; <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> ) or  ex <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">rl</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> ) )
 </span><b>by </b><i>, <a class="txt" href="jordan12.html#E2:40"><i><font color="Green">E43</font></i></a>, <a class="txt" href="jordan12.html#E4:40"><i><font color="Green">E54</font></i></a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_1"><b>suppose </b></a><a NAME="E1:40_1_1_1"/>
 ex <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">rl</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> &amp; <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:40_1_1_1"/>
( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">f</font> or <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">f</font> )
 <b>by </b><i><a class="ref" href="jordan12.html#T4" target="_self">Th4</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_2"><b>suppose </b></a><a NAME="E1:40_1_1_2"/>
 ex <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">rl</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:40_1_1_2"/>
( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">f</font> or <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">f</font> )
 <b>by </b><i><a class="ref" href="jordan12.html#T4" target="_self">Th4</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<div><i><font color="Green">E67</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:40_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">rp</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_2_1"/>
(  ex <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">rp</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> &amp; <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> ) or  ex <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">rp</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> ) )
 </span><b>by </b><i>, , <a class="txt" href="jordan12.html#E5:40"><i><font color="Green">E55</font></i></a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_1_1"><b>suppose </b></a><a NAME="E1:40_2_1_1"/>
 ex <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">rp</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> &amp; <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:40_2_1_1"/>
( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font> or <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font> )
 <b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_1_2"><b>suppose </b></a><a NAME="E1:40_2_1_2"/>
 ex <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">rp</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:40_2_1_2"/>
( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font> or <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font> )
 <b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<div><i><font color="Green">E69</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:40_3"/><i><font color="Green">E78</font></i>: 
not <font color="Maroon">rl</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_3_1"/>
(  ex <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">rl</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> &amp; <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> ) or  ex <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">rl</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> ) )
 </span><b>by </b><i>, <a class="txt" href="jordan12.html#E1"><i><font color="Green">Lemma40</font></i></a>, , <a class="txt" href="jordan12.html#E1:40"><i><font color="Green">E42</font></i></a>, <a class="txt" href="jordan12.html#E2:40"><i><font color="Green">E43</font></i></a>, <a class="txt" href="jordan12.html#E6:40"><i><font color="Green">E56</font></i></a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_3_1_1"><b>suppose </b></a><a NAME="E1:40_3_1_1"/>
 ex <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">rl</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> &amp; <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:40_3_1_1"/>
( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">f</font> or <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">f</font> )
 <b>by </b><i><a class="ref" href="jordan12.html#T4" target="_self">Th4</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_3_1_2"><b>suppose </b></a><a NAME="E1:40_3_1_2"/>
 ex <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">rl</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:40_3_1_2"/>
( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">f</font> or <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">f</font> )
 <b>by </b><i><a class="ref" href="jordan12.html#T4" target="_self">Th4</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<div><i><font color="Green">E79</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:40_4"/><i><font color="Green">E88</font></i>: 
not <font color="Maroon">rp</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_4_1"/>
(  ex <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">rp</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> &amp; <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> ) or  ex <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">rp</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> ) )
 </span><b>by </b><i>, <a class="txt" href="jordan12.html#E1"><i><font color="Green">Lemma40</font></i></a>, , <a class="txt" href="jordan12.html#E1:40"><i><font color="Green">E42</font></i></a>, , <a class="txt" href="jordan12.html#E7:40"><i><font color="Green">E57</font></i></a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_4_1_1"><b>suppose </b></a><a NAME="E1:40_4_1_1"/>
 ex <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">rp</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> &amp; <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:40_4_1_1"/>
( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font> or <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font> )
 <b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_4_1_2"><b>suppose </b></a><a NAME="E1:40_4_1_2"/>
 ex <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">rp</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:40_4_1_2"/>
( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font> or <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font> )
 <b>by </b><i><a class="ref" href="jordan12.html#T5" target="_self">Th5</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<div><i><font color="Green">E89</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:40_5"/><i><font color="Green">E90</font></i>: 
( not <font color="Maroon">rl</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp; not <font color="Maroon">rp</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_5_1"/>
( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">f</font> or <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">f</font> )
 </span><b>by </b><i><a class="txt" href="jordan12.html#E5:40"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan12.html#E9:40"><i><font color="Green">E58</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_5_1_1"><b>suppose </b></a><a NAME="E1:40_5_1_1"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_5_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_5_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_5_1_1_1_1"/>
( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font> or <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font> )
 </span><b>by </b><i><a class="txt" href="jordan12.html#E6:40"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan12.html#E9:40"><i><font color="Green">E58</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_5_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:40_5_1_1_1_1_1"/>
<font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><a NAME="E2:40_5_1_1_1_1_1"/><b>then </b>
 <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">f</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="jordan12.html#E10:40"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>hence </b><a NAME="E3:40_5_1_1_1_1_1"/>
for <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  holds <br/>(  not <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K3">`</a>  or  not <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> or  not <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 <b>by </b><i><a class="ref" href="gobrd14.html#T24">GOBRD14:24</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_5_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:40_5_1_1_1_1_2"/>
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:40_5_1_1_1_1_2"/>
for <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  holds <br/>(  not <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K3">`</a>  or  not <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> or  not <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 <b>by </b><i><a class="txt" href="jordan12.html#E10:40"><i><font color="Green">E59</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="E3:40_5_1_1"/>
for <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  holds <br/>(  not <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K3">`</a>  or  not <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> or  not <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_5_1_2"><b>suppose </b></a><a NAME="E1:40_5_1_2"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_5_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_5_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_5_1_2_1_1"/>
( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font> or <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font> )
 </span><b>by </b><i><a class="txt" href="jordan12.html#E6:40"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan12.html#E9:40"><i><font color="Green">E58</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_5_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:40_5_1_2_1_1_1"/>
<font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:40_5_1_2_1_1_1"/>
for <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  holds <br/>(  not <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K3">`</a>  or  not <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> or  not <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 <b>by </b><i><a class="txt" href="jordan12.html#E10:40"><i><font color="Green">E59</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_5_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:40_5_1_2_1_1_2"/>
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><a NAME="E2:40_5_1_2_1_1_2"/><b>then </b>
 <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">f</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="jordan12.html#E10:40"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>hence </b><a NAME="E3:40_5_1_2_1_1_2"/>
for <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  holds <br/>(  not <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K3">`</a>  or  not <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> or  not <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 <b>by </b><i><a class="ref" href="gobrd14.html#T24">GOBRD14:24</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:40_5_1_2"/>
for <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  holds <br/>(  not <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K3">`</a>  or  not <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> or  not <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<div><i><font color="Green">E93</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:40_6"/><i><font color="Green">E94</font></i>: 
( <font color="Maroon">rl</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp; <font color="Maroon">rp</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_6_1"/>
( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">f</font> or <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">f</font> )
 </span><b>by </b><i>, <a class="txt" href="jordan12.html#E10:40"><i><font color="Green">E59</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_6_1_1"><b>suppose </b></a><a NAME="E1:40_6_1_1"/><i><font color="Green">E139</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_6_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_6_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_6_1_1_1_1"/>
( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font> or <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font> )
 </span><b>by </b><i><a class="txt" href="jordan12.html#E4:40"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan12.html#E10:40"><i><font color="Green">E59</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_6_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:40_6_1_1_1_1_1"/>
<font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><a NAME="E2:40_6_1_1_1_1_1"/><b>then </b>
 <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">f</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="jordan12.html#E11:40"><i><font color="Green">E60</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>hence </b><a NAME="E3:40_6_1_1_1_1_1"/>
for <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  holds <br/>(  not <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K3">`</a>  or  not <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> or  not <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 <b>by </b><i><a class="ref" href="gobrd14.html#T24">GOBRD14:24</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_6_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:40_6_1_1_1_1_2"/>
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:40_6_1_1_1_1_2"/>
for <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  holds <br/>(  not <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K3">`</a>  or  not <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> or  not <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 <b>by </b><i><a class="txt" href="jordan12.html#E11:40"><i><font color="Green">E60</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="E3:40_6_1_1"/>
for <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  holds <br/>(  not <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K3">`</a>  or  not <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> or  not <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_6_1_2"><b>suppose </b></a><a NAME="E1:40_6_1_2"/><i><font color="Green">E143</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_6_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_6_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_6_1_2_1_1"/>
( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font> or <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font> )
 </span><b>by </b><i><a class="txt" href="jordan12.html#E4:40"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan12.html#E10:40"><i><font color="Green">E59</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_6_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:40_6_1_2_1_1_1"/>
<font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:40_6_1_2_1_1_1"/>
for <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  holds <br/>(  not <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K3">`</a>  or  not <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> or  not <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 <b>by </b><i><a class="txt" href="jordan12.html#E11:40"><i><font color="Green">E60</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_6_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:40_6_1_2_1_1_2"/>
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><a NAME="E2:40_6_1_2_1_1_2"/><b>then </b>
 <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">f</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="jordan12.html#E11:40"><i><font color="Green">E60</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>hence </b><a NAME="E3:40_6_1_2_1_1_2"/>
for <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  holds <br/>(  not <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K3">`</a>  or  not <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> or  not <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 <b>by </b><i><a class="ref" href="gobrd14.html#T24">GOBRD14:24</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:40_6_1_2"/>
for <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  holds <br/>(  not <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K3">`</a>  or  not <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> or  not <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<div><i><font color="Green">E144</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_7"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:40_7"/><i><font color="Green">E145</font></i>: 
( <font color="Maroon">rl</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp; not <font color="Maroon">rp</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_7_1"/>
( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">f</font> or <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">f</font> )
 </span><b>by </b><i>, <a class="txt" href="jordan12.html#E11:40"><i><font color="Green">E60</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_7_1_1"><b>suppose </b></a><a NAME="E1:40_7_1_1"/><i><font color="Green">E146</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_7_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_7_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_7_1_1_1_1"/>
( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font> or <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font> )
 </span><b>by </b><i><a class="txt" href="jordan12.html#E6:40"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan12.html#E11:40"><i><font color="Green">E60</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_7_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:40_7_1_1_1_1_1"/>
<font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><a NAME="E2:40_7_1_1_1_1_1"/><b>then </b>
 <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">f</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="jordan12.html#E12:40"><i><font color="Green">E61</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>hence </b><a NAME="E3:40_7_1_1_1_1_1"/>
for <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  holds <br/>(  not <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K3">`</a>  or  not <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> or  not <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 <b>by </b><i><a class="ref" href="gobrd14.html#T24">GOBRD14:24</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_7_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:40_7_1_1_1_1_2"/>
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:40_7_1_1_1_1_2"/>
for <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  holds <br/>(  not <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K3">`</a>  or  not <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> or  not <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 <b>by </b><i><a class="txt" href="jordan12.html#E12:40"><i><font color="Green">E61</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="E3:40_7_1_1"/>
for <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  holds <br/>(  not <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K3">`</a>  or  not <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> or  not <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_7_1_2"><b>suppose </b></a><a NAME="E1:40_7_1_2"/><i><font color="Green">E147</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_7_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_7_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_7_1_2_1_1"/>
( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font> or <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font> )
 </span><b>by </b><i><a class="txt" href="jordan12.html#E6:40"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan12.html#E11:40"><i><font color="Green">E60</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_7_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:40_7_1_2_1_1_1"/>
<font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:40_7_1_2_1_1_1"/>
for <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  holds <br/>(  not <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K3">`</a>  or  not <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> or  not <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 <b>by </b><i><a class="txt" href="jordan12.html#E12:40"><i><font color="Green">E61</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_7_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:40_7_1_2_1_1_2"/>
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><a NAME="E2:40_7_1_2_1_1_2"/><b>then </b>
 <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">f</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="jordan12.html#E12:40"><i><font color="Green">E61</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>hence </b><a NAME="E3:40_7_1_2_1_1_2"/>
for <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  holds <br/>(  not <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K3">`</a>  or  not <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> or  not <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 <b>by </b><i><a class="ref" href="gobrd14.html#T24">GOBRD14:24</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:40_7_1_2"/>
for <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  holds <br/>(  not <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K3">`</a>  or  not <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> or  not <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<div><i><font color="Green">E148</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_8"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:40_8"/><i><font color="Green">E149</font></i>: 
( not <font color="Maroon">rl</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp; <font color="Maroon">rp</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_8_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_8_1"/>
( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">f</font> or <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">f</font> )
 </span><b>by </b><i><a class="txt" href="jordan12.html#E5:40"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan12.html#E12:40"><i><font color="Green">E61</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_8_1_1"><b>suppose </b></a><a NAME="E1:40_8_1_1"/><i><font color="Green">E150</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_8_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_8_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_8_1_1_1_1"/>
( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font> or <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font> )
 </span><b>by </b><i><a class="txt" href="jordan12.html#E4:40"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan12.html#E12:40"><i><font color="Green">E61</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_8_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:40_8_1_1_1_1_1"/>
<font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><a NAME="E2:40_8_1_1_1_1_1"/><b>then </b>
 <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">f</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="jordan12.html#E13:40"><i><font color="Green">E62</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>hence </b><a NAME="E3:40_8_1_1_1_1_1"/>
for <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  holds <br/>(  not <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K3">`</a>  or  not <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> or  not <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 <b>by </b><i><a class="ref" href="gobrd14.html#T24">GOBRD14:24</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_8_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:40_8_1_1_1_1_2"/>
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:40_8_1_1_1_1_2"/>
for <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  holds <br/>(  not <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K3">`</a>  or  not <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> or  not <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 <b>by </b><i><a class="txt" href="jordan12.html#E13:40"><i><font color="Green">E62</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="E3:40_8_1_1"/>
for <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  holds <br/>(  not <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K3">`</a>  or  not <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> or  not <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_8_1_2"><b>suppose </b></a><a NAME="E1:40_8_1_2"/><i><font color="Green">E151</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_8_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_8_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_8_1_2_1_1"/>
( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font> or <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font> )
 </span><b>by </b><i><a class="txt" href="jordan12.html#E4:40"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan12.html#E12:40"><i><font color="Green">E61</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_8_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:40_8_1_2_1_1_1"/>
<font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:40_8_1_2_1_1_1"/>
for <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  holds <br/>(  not <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K3">`</a>  or  not <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> or  not <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 <b>by </b><i><a class="txt" href="jordan12.html#E13:40"><i><font color="Green">E62</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_8_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:40_8_1_2_1_1_2"/>
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><a NAME="E2:40_8_1_2_1_1_2"/><b>then </b>
 <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">f</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="jordan12.html#E13:40"><i><font color="Green">E62</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>hence </b><a NAME="E3:40_8_1_2_1_1_2"/>
for <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  holds <br/>(  not <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K3">`</a>  or  not <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> or  not <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 <b>by </b><i><a class="ref" href="gobrd14.html#T24">GOBRD14:24</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:40_8_1_2"/>
for <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  holds <br/>(  not <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K3">`</a>  or  not <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> or  not <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_9"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_9"/>
( <font color="Maroon">rl</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> or  not <font color="Maroon">rl</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_9_1"><b>suppose </b></a><a NAME="E1:40_9_1"/><i><font color="Green">E152</font></i>: 
<font color="Maroon">rl</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_9_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_9_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_9_1_1_1"/>
( <font color="Maroon">rp</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> or  not <font color="Maroon">rp</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_9_1_1_1_1"><b>suppose </b></a><a NAME="E1:40_9_1_1_1_1"/>
<font color="Maroon">rp</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:40_9_1_1_1_1"/>
for <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  holds <br/>(  not <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K3">`</a>  or  not <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> or  not <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 <b>by </b><i><a class="txt" href="jordan12.html#E9:40"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan12.html#E12:40"><i><font color="Green">E61</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_9_1_1_1_2"><b>suppose </b></a><a NAME="E1:40_9_1_1_1_2"/>
not <font color="Maroon">rp</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:40_9_1_1_1_2"/>
for <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  holds <br/>(  not <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K3">`</a>  or  not <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> or  not <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 <b>by </b><i><a class="txt" href="jordan12.html#E10:40"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan12.html#E12:40"><i><font color="Green">E61</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="E3:40_9_1"/>
for <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  holds <br/>(  not <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K3">`</a>  or  not <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> or  not <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_9_2"><b>suppose </b></a><a NAME="E1:40_9_2"/><i><font color="Green">E153</font></i>: 
not <font color="Maroon">rl</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_9_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_9_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_9_2_1_1"/>
( <font color="Maroon">rp</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> or  not <font color="Maroon">rp</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_9_2_1_1_1"><b>suppose </b></a><a NAME="E1:40_9_2_1_1_1"/>
<font color="Maroon">rp</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:40_9_2_1_1_1"/>
for <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  holds <br/>(  not <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K3">`</a>  or  not <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> or  not <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 <b>by </b><i><a class="txt" href="jordan12.html#E11:40"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan12.html#E12:40"><i><font color="Green">E61</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_9_2_1_1_2"><b>suppose </b></a><a NAME="E1:40_9_2_1_1_2"/>
not <font color="Maroon">rp</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:40_9_2_1_1_2"/>
for <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  holds <br/>(  not <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K3">`</a>  or  not <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> or  not <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 <b>by </b><i><a class="txt" href="jordan12.html#E7:40"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan12.html#E12:40"><i><font color="Green">E61</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="E3:40_9_2"/>
for <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  holds <br/>(  not <font color="Olive">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K3">`</a>  or  not <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> or  not <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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