<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font> be   <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="c2">C</font></span>)</span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="compact1.html#D6" target="_self" title="COMPACT1:def.6">COMPACT1:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">U</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c3">x</font> st <font color="Olive" title="b1">U</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> </span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">xx</font> = <font color="Maroon" title="c3">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T55" title="PRE_TOPC:th.55">PRE_TOPC:55</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">K</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c4">xx</font><span class="kw"> such that </span><br/><a NAME="E3:21_1_1"/><span class="lab"><font color="Green" title="E7">A1</font></span>: 
<font color="Maroon" title="c5">K</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="compact1.html#D6" target="_self" title="COMPACT1:def.6">Def6</a></span>;<br/>
<a NAME="E4:21_1_1"/><span class="lab"><font color="Green" title="E8">A2</font></span>: 
 <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="c2">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">KC</font> = <font color="Maroon" title="c5">K</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c2">C</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="c2">C</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">KC</font> = <font color="Maroon" title="c6">KC</font> as    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c3">x</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21_1_1"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="ref" href="topmetr.html#T6" title="TOPMETR:th.6">TOPMETR:6</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">KC</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">KC</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> </span><br/>

<a NAME="E7:21_1_1"/><span class="lab"><font color="Green" title="E9">A3</font></span>: 
<font color="Maroon" title="c4">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c5">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="connsp_2.html#D1" title="CONNSP_2:def.1">CONNSP_2:def 1</a></span>;<br/>
<a NAME="E8:21_1_1"/><span class="lab"><font color="Green" title="E10">A4</font></span>: 
 <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="c5">K</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">KK</font> = <font color="Maroon" title="c5">K</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c2">C</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="c5">K</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<a NAME="E10:21_1_1"/><span class="lab"><font color="Green" title="E11">A5</font></span>: 
<font color="Maroon" title="c8">KK</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21_1_1"><span class="lab"><font color="Green" title="E10">A4</font></span></a>, <a class="ref" href="pre_topc.html#T43" title="PRE_TOPC:th.43">PRE_TOPC:43</a></span>;<br/>
<a NAME="E11:21_1_1"/><span class="lab"><font color="Green" title="E12">A6</font></span>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c5">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_1.html#T44" title="TOPS_1:th.44">TOPS_1:44</a></span>;<br/>
<a NAME="E12:21_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A7</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">KC</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21_1_1"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E7:21_1_1"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E13:21_1_1"/>
<font color="Maroon" title="c1">X</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c5">K</font> is  <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_1_1"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E11:21_1_1"><span class="lab"><font color="Green" title="E12">A6</font></span></a>, <a class="txt" href="#E7:21_1_1"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="ref" href="compts_1.html#T12" title="COMPTS_1:th.12">COMPTS_1:12</a></span>;<br/>
<a NAME="E14:21_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">KK</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:21_1_1"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="ref" href="compts_1.html#T17" title="COMPTS_1:th.17">COMPTS_1:17</a></span>;<br/>
<a NAME="E15:21_1_1"/><span class="kw">then </span>
<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="c5">K</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">KK</font> is  <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:21_1_1"><span class="lab"><font color="Green" title="E13">A7</font></span></a>, <a class="ref" href="compts_1.html#T12" title="COMPTS_1:th.12">COMPTS_1:12</a></span>;<br/>
<a NAME="E16:21_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">X</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">K</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c2">C</font></span>)</span> is  <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T28" title="PRE_TOPC:th.28">PRE_TOPC:28</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<a NAME="E17:21_1_1"/><span class="kw">then </span>
<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="c2">C</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c7">KC</font> is  <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T28" title="PRE_TOPC:th.28">PRE_TOPC:28</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:21_1_1"/>
<font color="Maroon" title="c7">KC</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:21_1_1"><span class="lab"><font color="Green" title="E13">A7</font></span></a>, <a class="ref" href="compts_1.html#T12" title="COMPTS_1:th.12">COMPTS_1:12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
