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<b>let </b><font color="Maroon" title="c1">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">f</font> holds <br/> ex <font color="Olive" title="b3">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="b3">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="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Olive" title="b1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">C</font> &amp; <font color="Olive" title="b2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">C</font> )</span><br/><b>let </b><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <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> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">f</font> implies  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="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &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> &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> ) )</span><br/>





<b>assume </b><a NAME="E1:19"/><i><font color="Green" title="E18">A1</font></i>: 
 <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> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  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="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &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> &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> )</span><br/>

<a NAME="E2:19"/><i><font color="Green" title="E19">A2</font></i>: 
 <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> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 
<b>by </b><i><a class="ref" href="goboard9.html#T8" title="GOBOARD9:th.8">GOBOARD9:8</a></i>;<br/>
<a NAME="E3:19"/><i><font color="Green" title="E20">A3</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="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6" title="TOPREAL1:th.6">TOPREAL1:6</a></i>;<br/>
<a NAME="E4:19"/><i><font color="Green" title="E21">A4</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="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6" title="TOPREAL1:th.6">TOPREAL1:6</a></i>;<br/>
<a NAME="E5:19"/><b>then </b><i><font color="Green" title="E22">A5</font></i>: 
(  not <font color="Maroon" title="c3">p2</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="c1">f</font> or  not <font color="Maroon" title="c2">p1</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="c1">f</font> )
 
<b>by </b><i><a class="txt" href="#E1:19"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="#E2:19"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="#E3:19"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="jordan1j.html#T36" title="JORDAN1J:th.36">JORDAN1J:36</a></i>;<br/>
<a NAME="E6:19"/><i><font color="Green" title="E23">A6</font></i>: 
 <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c1">f</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="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<b>by </b><i><a class="ref" href="goboard9.html#D1" title="GOBOARD9:def.1">GOBOARD9:def 1</a></i>;<br/>
<a NAME="E7:19"/><i><font color="Green" title="E24">A7</font></i>: 
 <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</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="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<b>by </b><i><a class="ref" href="goboard9.html#D2" title="GOBOARD9:def.2">GOBOARD9:def 2</a></i>;<br/>
<a NAME="E8:19"/><i><font color="Green" title="E25">A8</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="c1">f</font>
 
<b>by </b><i><a class="txt" href="#E1:19"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="#E3:19"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E9:19"/><i><font color="Green" title="E26">A9</font></i>: 
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="c1">f</font>
 
<b>by </b><i><a class="txt" href="#E1:19"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="#E4:19"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">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" title="c2">p1</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="c1">f</font> or  not <font color="Maroon" title="c3">p2</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="c1">f</font> )
 </span><b>by </b><i><a class="txt" href="#E1:19"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="#E2:19"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="#E3:19"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="#E4:19"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="jordan1j.html#T36" title="JORDAN1J:th.36">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" title="E27">A10</font></i>: 
not <font color="Maroon" title="c2">p1</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="c1">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  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="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &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> &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> )</span><br/><div class="add"><a NAME="E2:19_1_1_1"/><b>then </b><i><font color="Green" title="E28">A11</font></i>: 
<font color="Maroon" title="c2">p1</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="c1">f</font>
 <b>by </b><i><a class="txt" href="#E8:19"><i><font color="Green" title="E25">A8</font></i></a>, <a class="ref" href="gobrd14.html#T27" title="GOBRD14:th.27">GOBRD14:27</a></i>;<br/><a NAME="E3:19_1_1_1"/>
<font color="Maroon" title="c3">p2</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="c1">f</font>
 <b>by </b><i><a class="txt" href="#E5:19"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="#E8:19"><i><font color="Green" title="E25">A8</font></i></a>, <a class="txt" href="#E9:19"><i><font color="Green" title="E26">A9</font></i></a>, <a class="txt" href="#E1:19_1_1_1"><i><font color="Green" title="E27">A10</font></i></a>, <a class="ref" href="gobrd14.html#T27" title="GOBRD14:th.27">GOBRD14:27</a></i>;<br/><b>hence </b><a NAME="E4:19_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="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &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> &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="#E6:19"><i><font color="Green" title="E23">A6</font></i></a>, <a class="txt" href="#E2:19_1_1_1"><i><font color="Green" title="E28">A11</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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" title="E27">A12</font></i>: 
not <font color="Maroon" title="c3">p2</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="c1">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  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="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &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> &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> )</span><br/><div class="add"><a NAME="E2:19_1_1_2"/><b>then </b><i><font color="Green" title="E28">A13</font></i>: 
<font color="Maroon" title="c3">p2</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="c1">f</font>
 <b>by </b><i><a class="txt" href="#E9:19"><i><font color="Green" title="E26">A9</font></i></a>, <a class="ref" href="gobrd14.html#T28" title="GOBRD14:th.28">GOBRD14:28</a></i>;<br/><a NAME="E3:19_1_1_2"/>
<font color="Maroon" title="c2">p1</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="c1">f</font>
 <b>by </b><i><a class="txt" href="#E5:19"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="#E8:19"><i><font color="Green" title="E25">A8</font></i></a>, <a class="txt" href="#E9:19"><i><font color="Green" title="E26">A9</font></i></a>, <a class="txt" href="#E1:19_1_1_2"><i><font color="Green" title="E27">A12</font></i></a>, <a class="ref" href="gobrd14.html#T28" title="GOBRD14:th.28">GOBRD14:28</a></i>;<br/><b>hence </b><a NAME="E4:19_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="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &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> &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="#E7:19"><i><font color="Green" title="E24">A7</font></i></a>, <a class="txt" href="#E2:19_1_1_2"><i><font color="Green" title="E28">A13</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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" 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="c1">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &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> &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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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