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

<div><b>hereby </b> <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">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>  ex <font color="Olive" title="b2">B</font> being   <a href="yellow13.html#M1" title="YELLOW13:mode.1">basis</a> of <font color="Olive" title="b1">x</font> st <font color="Olive" title="b2">B</font> is <a href="compact1.html#V1" title="COMPACT1:attr.1">compact</a> ) implies <font color="Maroon" title="c1">X</font> is <a href="waybel_3.html#V6" title="WAYBEL_3:attr.6">liminally-compact</a> )</span>
<div class="add"><b>assume </b><a NAME="E1:8_1_1_1"/><i><font color="Green" title="E3">A1</font></i>: 
<font color="Maroon" title="c1">X</font> is <a href="waybel_3.html#V6" title="WAYBEL_3:attr.6">liminally-compact</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">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>  ex <font color="Olive" title="b2">B</font> being   <a href="yellow13.html#M1" title="YELLOW13:mode.1">basis</a> of <font color="Olive" title="b1">x</font> st <font color="Olive" title="b2">B</font> is <a href="compact1.html#V1" title="COMPACT1:attr.1">compact</a></span><br/><b>let </b><font color="Maroon" title="c2">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</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">B</font> being   <a href="yellow13.html#M1" title="YELLOW13:mode.1">basis</a> of <font color="Maroon" title="c2">x</font> st <font color="Olive" title="b1">B</font> is <a href="compact1.html#V1" title="COMPACT1:attr.1">compact</a></span><br/><b>set </b><font color="Maroon" title="c3">B</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">Q</font> where <font color="Olive" title="b1">Q</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">x</font> : ( <font color="Olive" title="b1">Q</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> &amp;  ex <font color="Olive" title="b2">V</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">x</font> st <font color="Olive" title="b1">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">V</font> ) </span>}</span> ;<br/><a NAME="E2:8_1_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">Q</font> where <font color="Olive" title="b1">Q</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">x</font> : ( <font color="Olive" title="b1">Q</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> &amp;  ex <font color="Olive" title="b2">V</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">x</font> st <font color="Olive" title="b1">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">V</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">A</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">Q</font> where <font color="Olive" title="b1">Q</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">x</font> : ( <font color="Olive" title="b1">Q</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> &amp;  ex <font color="Olive" title="b2">V</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">x</font> st <font color="Olive" title="b1">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">V</font> ) </span>}</span>  or <font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</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> )</span><br/>

<b>assume </b><a NAME="E1:8_1_1_1_1"/>
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">Q</font> where <font color="Olive" title="b1">Q</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">x</font> : ( <font color="Olive" title="b1">Q</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> &amp;  ex <font color="Olive" title="b2">V</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">x</font> st <font color="Olive" title="b1">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">V</font> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</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></span><br/>

<a NAME="E2:8_1_1_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">Q</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">x</font> st <br/>( <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Q</font> &amp; <font color="Olive" title="b1">Q</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> &amp;  ex <font color="Olive" title="b2">V</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">x</font> st <font color="Olive" title="b1">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">V</font> )
 
;<br/>
<b>hence </b><a NAME="E3:8_1_1_1_1"/>
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</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 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><b>then reconsider </b><font color="Maroon" title="c4">B</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">Q</font> where <font color="Olive" title="b1">Q</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">x</font> : ( <font color="Olive" title="b1">Q</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> &amp;  ex <font color="Olive" title="b2">V</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">x</font> st <font color="Olive" title="b1">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">V</font> ) </span>}</span>  as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">X</font> ;<br/><a NAME="E4:8_1_1_1"/><i><font color="Green" title="E4">A2</font></i>: 
<font color="Maroon" title="c4">B</font> is    <a href="yellow13.html#M1" title="YELLOW13:mode.1">basis</a> of <font color="Maroon" title="c2">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">V</font> be    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">x</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow13.html#D2" title="YELLOW13:def.2">YELLOW13:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">x</font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">V</font> )</span><br/>

<a NAME="E1:8_1_1_1_2"/>
<font color="Maroon" title="c2">x</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">V</font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#D1" title="CONNSP_2:def.1">CONNSP_2:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">Q</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:8_1_1_1_2"/><i><font color="Green" title="E5">A3</font></i>: 
<font color="Maroon" title="c2">x</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="c6">Q</font>
 <b>and </b><br/><a NAME="E4:8_1_1_1_2"/><i><font color="Green" title="E6">A4</font></i>: 
<font color="Maroon" title="c6">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c5">V</font>
 <b>and </b><br/><a NAME="E5:8_1_1_1_2"/><i><font color="Green" title="E7">A5</font></i>: 
<font color="Maroon" title="c6">Q</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a>
 <b>by </b><i><a class="txt" href="compact1.html#E1:8_1_1_1"><i><font color="Green" title="E3">A1</font></i></a>, <a class="ref" href="waybel_3.html#D9" title="WAYBEL_3:def.9">WAYBEL_3:def 9</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">Q</font> = <font color="Maroon" title="c6">Q</font> as    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">x</font> <b>by </b><i><a class="txt" href="compact1.html#E3:8_1_1_1_2"><i><font color="Green" title="E5">A3</font></i></a>, <a class="ref" href="connsp_2.html#D1" title="CONNSP_2:def.1">CONNSP_2:def 1</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">Q</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c7">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">V</font> )</span><br/>

<a NAME="E7:8_1_1_1_2"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c5">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">V</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T44" title="TOPS_1:th.44">TOPS_1:44</a></i>;<br/>
<b>hence </b><a NAME="E8:8_1_1_1_2"/>
( <font color="Maroon" title="c7">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c7">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">V</font> )
 <b>by </b><i><a class="txt" href="compact1.html#E4:8_1_1_1_2"><i><font color="Green" title="E6">A4</font></i></a>, <a class="txt" href="compact1.html#E5:8_1_1_1_2"><i><font color="Green" title="E7">A5</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</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><a NAME="E5:8_1_1_1"/>
<font color="Maroon" title="c4">B</font> is <a href="compact1.html#V1" title="COMPACT1:attr.1">compact</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">U</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="compact1.html#D1" target="_self" title="COMPACT1:def.1">COMPACT1:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">U</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> implies <font color="Maroon" title="c5">U</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> )</span><br/>

<b>assume </b><a NAME="E1:8_1_1_1_3"/>
<font color="Maroon" title="c5">U</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">U</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a></span><br/>

<a NAME="E2:8_1_1_1_3"/><b>then </b>
 ex <font color="Olive" title="b1">Q</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">x</font> st <br/>( <font color="Maroon" title="c5">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Q</font> &amp; <font color="Olive" title="b1">Q</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> &amp;  ex <font color="Olive" title="b2">V</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">x</font> st <font color="Olive" title="b1">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">V</font> )
 
;<br/>
<b>hence </b><a NAME="E3:8_1_1_1_3"/>
<font color="Maroon" title="c5">U</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a>
 ; <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><b>hence </b><a NAME="E6:8_1_1_1"/>
 ex <font color="Olive" title="b1">B</font> being   <a href="yellow13.html#M1" title="YELLOW13:mode.1">basis</a> of <font color="Maroon" title="c2">x</font> st <font color="Olive" title="b1">B</font> is <a href="compact1.html#V1" title="COMPACT1:attr.1">compact</a>
 <b>by </b><i><a class="txt" href="compact1.html#E4:8_1_1_1"><i><font color="Green" title="E4">A2</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>

<b>assume </b><a NAME="E2:8_1_1"/><i><font color="Green" title="E3">A6</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>  ex <font color="Olive" title="b2">B</font> being   <a href="yellow13.html#M1" title="YELLOW13:mode.1">basis</a> of <font color="Olive" title="b1">x</font> st <font color="Olive" title="b2">B</font> is <a href="compact1.html#V1" title="COMPACT1:attr.1">compact</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">X</font> is <a href="waybel_3.html#V6" title="WAYBEL_3:attr.6">liminally-compact</a></span><br/>

<b>let </b><font color="Maroon" title="c2">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_3.html#D9" title="WAYBEL_3:def.9">WAYBEL_3:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>  holds <br/>(  not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>1</sub></font> or  not <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> or  ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Maroon" title="c2">x</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="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> ) )</span><br/><b>let </b><font color="Maroon" title="c3">P</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> or  not <font color="Maroon" title="c3">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> or  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Maroon" title="c2">x</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="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> ) )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E3:8_1_1"/><i><font color="Green" title="E4">A7</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 <b>and </b><br/><a NAME="E4:8_1_1"/><i><font color="Green" title="E5">A8</font></i>: 
<font color="Maroon" title="c3">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Maroon" title="c2">x</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="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> )</span><br/>

<b>consider </b><font color="Maroon" title="c4">B</font> being    <a href="yellow13.html#M1" title="YELLOW13:mode.1">basis</a> of <font color="Maroon" title="c2">x</font><b> such that </b><br/><a NAME="E6:8_1_1"/><i><font color="Green" title="E6">A9</font></i>: 
<font color="Maroon" title="c4">B</font> is <a href="compact1.html#V1" title="COMPACT1:attr.1">compact</a>
 <b>by </b><i><a class="txt" href="compact1.html#E2:8_1_1"><i><font color="Green" title="E3">A6</font></i></a></i>;<br/>
<a NAME="E7:8_1_1"/>
<font color="Maroon" title="c2">x</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="c3">P</font>
 
<b>by </b><i><a class="txt" href="compact1.html#E3:8_1_1"><i><font color="Green" title="E4">A7</font></i></a>, <a class="txt" href="compact1.html#E4:8_1_1"><i><font color="Green" title="E5">A8</font></i></a>, <a class="ref" href="tops_1.html#T55" title="TOPS_1:th.55">TOPS_1:55</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">Q</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E9:8_1_1"/><i><font color="Green" title="E7">A10</font></i>: 
<font color="Maroon" title="c5">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 <b>and </b><br/><a NAME="E10:8_1_1"/><i><font color="Green" title="E8">A11</font></i>: 
( <font color="Maroon" title="c2">x</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">Q</font> &amp; <font color="Maroon" title="c5">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">P</font> )
 <b>by </b><i><a class="ref" href="yellow13.html#D1" title="YELLOW13:def.1">YELLOW13:def 1</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">Q</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">x</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">Q</font> &amp; <font color="Maroon" title="c5">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c5">Q</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> )</span><br/>

<b>thus </b><a NAME="E11:8_1_1"/>
( <font color="Maroon" title="c2">x</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">Q</font> &amp; <font color="Maroon" title="c5">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">P</font> )
 <b>by </b><i><a class="txt" href="compact1.html#E10:8_1_1"><i><font color="Green" title="E8">A11</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">Q</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a></span><br/>

<b>thus </b><a NAME="E12:8_1_1"/>
<font color="Maroon" title="c5">Q</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a>
 <b>by </b><i><a class="txt" href="compact1.html#E6:8_1_1"><i><font color="Green" title="E6">A9</font></i></a>, <a class="txt" href="compact1.html#E9:8_1_1"><i><font color="Green" title="E7">A10</font></i></a>, <a class="ref" href="compact1.html#D1" target="_self" title="COMPACT1:def.1">Def1</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>
