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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">X</font> is  <a href="connsp_2.html#V1" title="CONNSP_2:attr.1">locally_connected</a>  &amp; <font color="Maroon" title="c1">X</font> is  <a href="pre_topc.html#V10" title="PRE_TOPC:attr.10">normal</a>  implies  for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b2">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b1">A</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  &amp; <font color="Olive" title="b2">B</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  &amp; <font color="Olive" title="b1">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">B</font> &amp; <font color="Olive" title="b1">A</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  &amp; <font color="Olive" title="b2">B</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  holds <br/> ex <font color="Olive" title="b3">R</font>, <font color="Olive" title="b4">S</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b3">R</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  &amp; <font color="Olive" title="b4">S</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  &amp; <font color="Olive" title="b3">R</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Olive" title="b4">S</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">R</font> &amp; <font color="Olive" title="b2">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b4">S</font> &amp; <font color="Olive" title="b3">R</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b4">S</font> ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:28"/><span class="lab"><font color="Green" title="E22">A1</font></span>: 
<font color="Maroon" title="c1">X</font> is  <a href="connsp_2.html#V1" title="CONNSP_2:attr.1">locally_connected</a> 
 <span class="kw">and </span><br/><a NAME="E2:28"/><span class="lab"><font color="Green" title="E23">A2</font></span>: 
<font color="Maroon" title="c1">X</font> is  <a href="pre_topc.html#V10" title="PRE_TOPC:attr.10">normal</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b2">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b1">A</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  &amp; <font color="Olive" title="b2">B</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  &amp; <font color="Olive" title="b1">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">B</font> &amp; <font color="Olive" title="b1">A</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  &amp; <font color="Olive" title="b2">B</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  holds <br/> ex <font color="Olive" title="b3">R</font>, <font color="Olive" title="b4">S</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b3">R</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  &amp; <font color="Olive" title="b4">S</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  &amp; <font color="Olive" title="b3">R</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Olive" title="b4">S</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">R</font> &amp; <font color="Olive" title="b2">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b4">S</font> &amp; <font color="Olive" title="b3">R</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b4">S</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c3">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c2">A</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  &amp; <font color="Maroon" title="c3">B</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  &amp; <font color="Maroon" title="c2">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c2">A</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  &amp; <font color="Maroon" title="c3">B</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  implies  ex <font color="Olive" title="b1">R</font>, <font color="Olive" title="b2">S</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">R</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  &amp; <font color="Olive" title="b2">S</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  &amp; <font color="Olive" title="b1">R</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Olive" title="b2">S</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">R</font> &amp; <font color="Maroon" title="c3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">S</font> &amp; <font color="Olive" title="b1">R</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">S</font> ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:28"/><span class="lab"><font color="Green" title="E24">A3</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">and </span><br/><a NAME="E4:28"/><span class="lab"><font color="Green" title="E25">A4</font></span>: 
<font color="Maroon" title="c3">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">and </span><br/><a NAME="E5:28"/><span class="lab"><font color="Green" title="E26">A5</font></span>: 
<font color="Maroon" title="c2">A</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 <span class="kw">and </span><br/><a NAME="E6:28"/><span class="lab"><font color="Green" title="E27">A6</font></span>: 
( <font color="Maroon" title="c3">B</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  &amp; <font color="Maroon" title="c2">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">B</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c2">A</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  or  not <font color="Maroon" title="c3">B</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  or  ex <font color="Olive" title="b1">R</font>, <font color="Olive" title="b2">S</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">R</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  &amp; <font color="Olive" title="b2">S</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  &amp; <font color="Olive" title="b1">R</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Olive" title="b2">S</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">R</font> &amp; <font color="Maroon" title="c3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">S</font> &amp; <font color="Olive" title="b1">R</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">S</font> ) )</span><br/>

<a NAME="E7:28"/>
<font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">B</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T3" title="PRE_TOPC:th.3">PRE_TOPC:3</a></span>;<br/>
<a NAME="E8:28"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A7</font></span>: 
<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="ref" href="pre_topc.html#T4" title="PRE_TOPC:th.4">PRE_TOPC:4</a></span>;<br/>
<a NAME="E9:28"/>
<font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28"><span class="lab"><font color="Green" title="E24">A3</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E11:28"/><span class="lab"><font color="Green" title="E29">A8</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T12" title="PRE_TOPC:th.12">PRE_TOPC:12</a></span>;<br/>
<a NAME="E12:28"/>
( <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c3">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:28"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="ref" href="subset_1.html#T23" title="SUBSET_1:th.23">SUBSET_1:23</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">G</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E14:28"/><span class="lab"><font color="Green" title="E30">A9</font></span>: 
<font color="Maroon" title="c5">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">and </span><br/><a NAME="E15:28"/><span class="lab"><font color="Green" title="E31">A10</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font>
 <span class="kw">and </span><br/><a NAME="E16:28"/><span class="lab"><font color="Green" title="E32">A11</font></span>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c5">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E3:28"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E5:28"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="txt" href="#E8:28"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="ref" href="connsp_2.html#T20" target="_self" title="CONNSP_2:th.20">Th26</a></span>;<br/>
<a NAME="E17:28"/><span class="lab"><font color="Green" title="E33">A12</font></span>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c5">G</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:28"><span class="lab"><font color="Green" title="E32">A11</font></span></a>, <a class="ref" href="subset_1.html#T23" title="SUBSET_1:th.23">SUBSET_1:23</a></span>;<br/>
<a NAME="E18:28"/><span class="lab"><font color="Green" title="E34">A13</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:28"><span class="lab"><font color="Green" title="E31">A10</font></span></a>, <a class="txt" href="#E11:28"><span class="lab"><font color="Green" title="E29">A8</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">G</font> = <font color="Maroon" title="c5">G</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E15:28"><span class="lab"><font color="Green" title="E31">A10</font></span></a></span>;<br/>
<a NAME="E20:28"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">G</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:28"><span class="lab"><font color="Green" title="E34">A13</font></span></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">y</font> = <font color="Maroon" title="c4">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">G</font></span>)</span> ;<br/>
<a NAME="E22:28"/><span class="lab"><font color="Green" title="E35">A14</font></span>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c6">G</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></span>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E23:28"/><span class="lab"><font color="Green" title="E36">A15</font></span>: 
<font color="Maroon" title="c2">A</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> 
 <span class="kw">and </span><br/><a NAME="E24:28"/><span class="lab"><font color="Green" title="E37">A16</font></span>: 
<font color="Maroon" title="c3">B</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">R</font>, <font color="Olive" title="b2">S</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">R</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  &amp; <font color="Olive" title="b2">S</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  &amp; <font color="Olive" title="b1">R</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Olive" title="b2">S</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">R</font> &amp; <font color="Maroon" title="c3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">S</font> &amp; <font color="Olive" title="b1">R</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">S</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c8">H</font> =  <a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <font color="Maroon" title="c7">y</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">H1</font> =  <a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <font color="Maroon" title="c7">y</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T11" title="PRE_TOPC:th.11">PRE_TOPC:11</a></span>;<br/>
<span class="kw">take </span><font color="Maroon" title="c10">R</font> = <font color="Maroon" title="c9">H1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">S</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Maroon" title="c10">R</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  &amp; <font color="Olive" title="b1">S</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  &amp; <font color="Maroon" title="c10">R</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Olive" title="b1">S</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">R</font> &amp; <font color="Maroon" title="c3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S</font> &amp; <font color="Maroon" title="c10">R</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b1">S</font> )</span><br/>

<a NAME="E26:28"/><span class="lab"><font color="Green" title="E38">A17</font></span>: 
 <a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <font color="Maroon" title="c7">y</font> is  <a href="connsp_1.html#V3" title="CONNSP_1:attr.3">a_component</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="connsp_1.html#T40" title="CONNSP_1:th.40">CONNSP_1:40</a></span>;<br/>
<a NAME="E27:28"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A18</font></span>: 
<font color="Maroon" title="c9">H1</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <font color="Maroon" title="c6">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="connsp_1.html#D6" title="CONNSP_1:def.6">CONNSP_1:def 6</a></span>;<br/>
<a NAME="E28:28"/>
<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">G</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:28"><span class="lab"><font color="Green" title="E31">A10</font></span></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">A1</font> = <font color="Maroon" title="c2">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">G</font></span>)</span> ;<br/>
<a NAME="E30:28"/><span class="lab"><font color="Green" title="E40">A19</font></span>: 
(  <a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <font color="Maroon" title="c7">y</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <font color="Maroon" title="c7">y</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="connsp_1.html#T38" title="CONNSP_1:th.38">CONNSP_1:38</a></span>;<br/>
<a NAME="E31:28"/>
<font color="Maroon" title="c11">A1</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:28"><span class="lab"><font color="Green" title="E36">A15</font></span></a>, <a class="ref" href="connsp_1.html#T23" title="CONNSP_1:th.23">CONNSP_1:23</a></span>;<br/>
<a NAME="E32:28"/><span class="kw">then </span>
( <font color="Maroon" title="c11">A1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <font color="Maroon" title="c7">y</font> or <font color="Maroon" title="c11">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <font color="Maroon" title="c7">y</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:28"><span class="lab"><font color="Green" title="E38">A17</font></span></a>, <a class="ref" href="connsp_1.html#T36" title="CONNSP_1:th.36">CONNSP_1:36</a></span>;<br/>
<a NAME="E33:28"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A20</font></span>: 
( <font color="Maroon" title="c11">A1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <font color="Maroon" title="c7">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c11">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <font color="Maroon" title="c7">y</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/>
<a NAME="E34:28"/>
 <a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <font color="Maroon" title="c7">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">G</font></span>)</span>
 
;<br/>
<a NAME="E35:28"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A21</font></span>: 
<font color="Maroon" title="c10">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>;<br/>
<a NAME="E36:28"/>
<font color="Maroon" title="c6">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c6">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T18" title="PRE_TOPC:th.18">PRE_TOPC:18</a></span>;<br/>
<a NAME="E37:28"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A22</font></span>: 
<font color="Maroon" title="c10">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c6">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:28"><span class="lab"><font color="Green" title="E42">A21</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E38:28"/>
<font color="Maroon" title="c3">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28"><span class="lab"><font color="Green" title="E25">A4</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">z</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E40:28"/><span class="lab"><font color="Green" title="E44">A23</font></span>: 
<font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T12" title="PRE_TOPC:th.12">PRE_TOPC:12</a></span>;<br/>
<a NAME="E41:28"/><span class="lab"><font color="Green" title="E45">A24</font></span>: 
<font color="Maroon" title="c3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:28"><span class="lab"><font color="Green" title="E33">A12</font></span></a>, <a class="ref" href="subset_1.html#T23" title="SUBSET_1:th.23">SUBSET_1:23</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">C</font> = <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:28"><span class="lab"><font color="Green" title="E44">A23</font></span></a></span>;<br/>
<a NAME="E43:28"/>
<font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:28"><span class="lab"><font color="Green" title="E44">A23</font></span></a>, <a class="txt" href="#E41:28"><span class="lab"><font color="Green" title="E45">A24</font></span></a></span>;<br/>
<a NAME="E44:28"/><span class="kw">then </span>
<font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c13">C</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">z1</font> = <font color="Maroon" title="c12">z</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c13">C</font></span>)</span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">V</font> =  <a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <font color="Maroon" title="c14">z1</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">V1</font> =  <a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <font color="Maroon" title="c14">z1</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T11" title="PRE_TOPC:th.11">PRE_TOPC:11</a></span>;<br/>
<span class="kw">take </span><font color="Maroon" title="c17">S</font> = <font color="Maroon" title="c16">V1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">R</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  &amp; <font color="Maroon" title="c17">S</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  &amp; <font color="Maroon" title="c10">R</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c17">S</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">R</font> &amp; <font color="Maroon" title="c3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">S</font> &amp; <font color="Maroon" title="c10">R</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c17">S</font> )</span><br/>

<a NAME="E47:28"/><span class="lab"><font color="Green" title="E46">A25</font></span>: 
 <a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <font color="Maroon" title="c14">z1</font> is  <a href="connsp_1.html#V3" title="CONNSP_1:attr.3">a_component</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="connsp_1.html#T40" title="CONNSP_1:th.40">CONNSP_1:40</a></span>;<br/>
<a NAME="E48:28"/>
<font color="Maroon" title="c3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:28"><span class="lab"><font color="Green" title="E45">A24</font></span></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c18">B1</font> = <font color="Maroon" title="c3">B</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span> ;<br/>
<a NAME="E50:28"/><span class="lab"><font color="Green" title="E47">A26</font></span>: 
(  <a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <font color="Maroon" title="c14">z1</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  &amp; <font color="Maroon" title="c14">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <font color="Maroon" title="c14">z1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="connsp_1.html#T38" title="CONNSP_1:th.38">CONNSP_1:38</a></span>;<br/>
<a NAME="E51:28"/>
<font color="Maroon" title="c18">B1</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:28"><span class="lab"><font color="Green" title="E37">A16</font></span></a>, <a class="ref" href="connsp_1.html#T23" title="CONNSP_1:th.23">CONNSP_1:23</a></span>;<br/>
<a NAME="E52:28"/><span class="kw">then </span>
( <font color="Maroon" title="c18">B1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <font color="Maroon" title="c14">z1</font> or <font color="Maroon" title="c18">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <font color="Maroon" title="c14">z1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:28"><span class="lab"><font color="Green" title="E46">A25</font></span></a>, <a class="ref" href="connsp_1.html#T36" title="CONNSP_1:th.36">CONNSP_1:36</a></span>;<br/>
<a NAME="E53:28"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A27</font></span>: 
( <font color="Maroon" title="c18">B1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <font color="Maroon" title="c14">z1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c18">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <font color="Maroon" title="c14">z1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/>
<a NAME="E54:28"/>
 <a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <font color="Maroon" title="c14">z1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span>
 
;<br/>
<a NAME="E55:28"/><span class="kw">then </span>
<font color="Maroon" title="c17">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>;<br/>
<a NAME="E56:28"/><span class="kw">then </span>
<font color="Maroon" title="c10">R</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c17">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:28"><span class="lab"><font color="Green" title="E43">A22</font></span></a>, <a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">XBOOLE_1:27</a></span>;<br/>
<a NAME="E57:28"/><span class="kw">then </span>
<font color="Maroon" title="c10">R</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c17">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:28"><span class="lab"><font color="Green" title="E35">A14</font></span></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/>
<a NAME="E58:28"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A28</font></span>: 
<font color="Maroon" title="c10">R</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c17">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
;<br/>
<a NAME="E59:28"/>
<font color="Maroon" title="c16">V1</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:28"><span class="lab"><font color="Green" title="E46">A25</font></span></a>, <a class="ref" href="connsp_1.html#D6" title="CONNSP_1:def.6">CONNSP_1:def 6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E60:28"/>
( <font color="Maroon" title="c10">R</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  &amp; <font color="Maroon" title="c17">S</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  &amp; <font color="Maroon" title="c10">R</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c17">S</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">R</font> &amp; <font color="Maroon" title="c3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">S</font> &amp; <font color="Maroon" title="c10">R</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c17">S</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E14:28"><span class="lab"><font color="Green" title="E30">A9</font></span></a>, <a class="txt" href="#E11:28"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="txt" href="#E27:28"><span class="lab"><font color="Green" title="E39">A18</font></span></a>, <a class="txt" href="#E33:28"><span class="lab"><font color="Green" title="E41">A20</font></span></a>, <a class="txt" href="#E30:28"><span class="lab"><font color="Green" title="E40">A19</font></span></a>, <a class="txt" href="#E40:28"><span class="lab"><font color="Green" title="E44">A23</font></span></a>, <a class="txt" href="#E53:28"><span class="lab"><font color="Green" title="E48">A27</font></span></a>, <a class="txt" href="#E50:28"><span class="lab"><font color="Green" title="E47">A26</font></span></a>, <a class="txt" href="#E58:28"><span class="lab"><font color="Green" title="E49">A28</font></span></a>, <a class="ref" href="connsp_2.html#T18" target="_self" title="CONNSP_2:th.18">Th24</a>, <a class="ref" href="connsp_1.html#T23" title="CONNSP_1:th.23">CONNSP_1:23</a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
