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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>assume </b><a NAME="E1:19"/><i><font color="Green">E42</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E2:19"/><i><font color="Green">E43</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i><a class="ref" href="goboard9.html#T8">GOBOARD9:8</a></i>;<br/>
<a NAME="E3:19"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">p1</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>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>
<a NAME="E4:19"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">p2</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>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>
<a NAME="E5:19"/><b>then </b><i><font color="Green">E55</font></i>: 
(  not <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font> or  not <font color="Maroon">p1</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="jordan1j.html#T36">JORDAN1J:36</a></i>;<br/>
<a NAME="E6:19"/><i><font color="Green">E56</font></i>: 
 <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">f</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> 
 
<b>by </b><i><a class="ref" href="goboard9.html#D1">GOBOARD9:def 1</a></i>;<br/>
<a NAME="E7:19"/><i><font color="Green">E57</font></i>: 
 <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</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> 
 
<b>by </b><i><a class="ref" href="goboard9.html#D2">GOBOARD9:def 2</a></i>;<br/>
<a NAME="E8:19"/><i><font color="Green">E58</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>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E9:19"/><i><font color="Green">E59</font></i>: 
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>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_1_1"/>
(  not <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  not <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="ref" href="jordan1j.html#T36">JORDAN1J:36</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_1_1"><b>suppose </b></a><a NAME="E1:19_1_1_1"/><i><font color="Green">E60</font></i>: 
not <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="E2:19_1_1_1"/><b>then </b><i><font color="Green">E61</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>
 <b>by </b><i>, <a class="ref" href="gobrd14.html#T27">GOBRD14:27</a></i>;<br/><a NAME="E3:19_1_1_1"/>
<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="gobrd14.html#T27">GOBRD14:27</a></i>;<br/><b>hence </b><a NAME="E4:19_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">p1</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> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_1_2"><b>suppose </b></a><a NAME="E1:19_1_1_2"/><i><font color="Green">E62</font></i>: 
not <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"><a NAME="E2:19_1_1_2"/><b>then </b><i><font color="Green">E63</font></i>: 
<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="gobrd14.html#T28">GOBRD14:28</a></i>;<br/><a NAME="E3:19_1_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>
 <b>by </b><i>, , , , <a class="ref" href="gobrd14.html#T28">GOBRD14:28</a></i>;<br/><b>hence </b><a NAME="E4:19_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">p1</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> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E11:19"/>
 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">p1</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/>


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