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

<b>let </b><font color="Maroon" title="c1">T</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">T</font> is <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a> implies for <font color="Olive" title="b1">N</font> being  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">T</font>  ex <font color="Olive" title="b2">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> st <font color="Olive" title="b2">x</font> <a href="waybel_9.html#R3" title="WAYBEL_9:pred.3">is_a_cluster_point_of</a> <font color="Olive" title="b1">N</font> )</span><br/>

<b>assume </b><a NAME="E1:39"/><i><font color="Green" title="E25">A1</font></i>: 
<font color="Maroon" title="c1">T</font> is <a href="compts_1.html#V1" title="COMPTS_1:attr.1">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">N</font> being  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">T</font>  ex <font color="Olive" title="b2">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> st <font color="Olive" title="b2">x</font> <a href="waybel_9.html#R3" title="WAYBEL_9:pred.3">is_a_cluster_point_of</a> <font color="Olive" title="b1">N</font></span><br/>

<b>let </b><font color="Maroon" title="c2">N</font> be   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">T</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">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> st <font color="Olive" title="b1">x</font> <a href="waybel_9.html#R3" title="WAYBEL_9:pred.3">is_a_cluster_point_of</a> <font color="Maroon" title="c2">N</font></span><br/>

<b>set </b><font color="Maroon" title="c3">F</font> = <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="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is    <a href="yellow19.html#M1" title="YELLOW19:mode.1">Subset</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">N</font> : verum </span>}</span> ;<br/>
<a NAME="E2:39"/>
<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="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is    <a href="yellow19.html#M1" title="YELLOW19:mode.1">Subset</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">N</font> : verum </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> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is    <a href="yellow19.html#M1" title="YELLOW19:mode.1">Subset</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">N</font> : verum </span>}</span>  or <font color="Maroon" title="c4">x</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> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> )</span><br/>

<b>assume </b><a NAME="E1:39_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is    <a href="yellow19.html#M1" title="YELLOW19:mode.1">Subset</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">N</font> : verum </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">x</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> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><br/>

<a NAME="E2:39_1"/><b>then </b>
 ex <font color="Olive" title="b1">A</font> being   <a href="yellow19.html#M1" title="YELLOW19:mode.1">Subset</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">N</font> st <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b1">A</font>
 
;<br/>
<b>hence </b><a NAME="E3:39_1"/>
<font color="Maroon" title="c4">x</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> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 ; <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">F</font> = <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="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is    <a href="yellow19.html#M1" title="YELLOW19:mode.1">Subset</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">N</font> : verum </span>}</span>  as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<b>consider </b><font color="Maroon" title="c5">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c4">F</font>;<br/>
<a NAME="E5:39"/><i><font color="Green" title="E26">A2</font></i>: 
<font color="Maroon" title="c4">F</font> is <a href="tops_2.html#V2" title="TOPS_2:attr.2">closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">S</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D2" title="TOPS_2:def.2">TOPS_2:def 2</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="c6">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font> or <font color="Maroon" title="c6">S</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )</span><br/>

<b>assume </b><a NAME="E1:39_2"/>
<font color="Maroon" title="c6">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">S</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a></span><br/>

<a NAME="E2:39_2"/><b>then </b>
 ex <font color="Olive" title="b1">A</font> being   <a href="yellow19.html#M1" title="YELLOW19:mode.1">Subset</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">N</font> st <font color="Maroon" title="c6">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b1">A</font>
 
;<br/>
<b>hence </b><a NAME="E3:39_2"/>
<font color="Maroon" title="c6">S</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</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>
<a NAME="E6:39"/>
<font color="Maroon" title="c4">F</font> is <a href="finset_1.html#V3" title="FINSET_1:attr.3">centered</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_3"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c6">A0</font> being    <a href="yellow19.html#M1" title="YELLOW19:mode.1">Subset</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">N</font>;<br/>
<a NAME="E2:39_3"/>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c6">A0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font>
 
;<br/>
<b>hence </b><a NAME="E3:39_3"/>
<font color="Maroon" title="c4">F</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> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="finset_1.html#D3" title="FINSET_1:def.3">FINSET_1:def 3</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="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  not <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">F</font> or  not <font color="Olive">b<sub>1</sub></font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> or  not  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/>

<b>let </b><font color="Maroon" title="c7">G</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  not <font color="Maroon" title="c7">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">F</font> or  not <font color="Maroon" title="c7">G</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> or  not  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c7">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/>

<b>assume </b><a NAME="E4:39_3"/><i><font color="Green" title="E27">A3</font></i>: 
( <font color="Maroon" title="c7">G</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="c7">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">F</font> &amp; <font color="Maroon" title="c7">G</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c7">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">A</font> being   <a href="yellow19.html#M1" title="YELLOW19:mode.1">Subset</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">N</font> ex <font color="Olive" title="b2">i</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b1">A</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Olive" title="b2">i</font></span>)</span> );<br/>
<div><i><font color="Green" title="E28">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">G</font> implies  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">N</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">x</font>,<font color="Olive" title="b1">y</font>] ) )</span><br/><b>assume </b><a NAME="E1:39_3_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">G</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">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">N</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">x</font>,<font color="Olive" title="b1">y</font>] )</span><br/><a NAME="E2:39_3_1"/><b>then </b>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font>
 <b>by </b><i><a class="txt" href="#E4:39_3"><i><font color="Green" title="E27">A3</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c9">A</font> being    <a href="yellow19.html#M1" title="YELLOW19:mode.1">Subset</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">N</font><b> such that </b><br/><a NAME="E4:39_3_1"/><i><font color="Green" title="E29">A5</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c9">A</font>
 ;<br/><b>consider </b><font color="Maroon" title="c10">i</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font><b> such that </b><br/><a NAME="E6:39_3_1"/><i><font color="Green" title="E30">A6</font></i>: 
<font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c10">i</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow19.html#D2" target="_self" title="YELLOW19:def.2">Def2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">y</font> = <font color="Maroon" title="c10">i</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><b>take </b><font color="Maroon" title="c12">y</font> = <font color="Maroon" title="c11">y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">N</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c12">y</font>] )</span><br/><b>thus </b><a NAME="E8:39_3_1"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">N</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c12">y</font>]</span><br/><b>thus </b><a NAME="E9:39_3_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c12">y</font>]
 <b>by </b><i><a class="txt" href="#E4:39_3_1"><i><font color="Green" title="E29">A5</font></i></a>, <a class="txt" href="#E6:39_3_1"><i><font color="Green" title="E30">A6</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>consider </b><font color="Maroon" title="c8">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E7:39_3"/><i><font color="Green" title="E29">A7</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">G</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">N</font> )
 <b>and </b><br/><a NAME="E8:39_3"/><i><font color="Green" title="E30">A8</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">G</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="wellord2.html#S1" title="WELLORD2:sch.1">WELLORD2:sch 1</a>(<a class="txt" href="#E5:39_3"><i><font color="Green" title="E28">A4</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c9">B</font> =  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">f</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">N</font> <b>by </b><i><a class="txt" href="#E4:39_3"><i><font color="Green" title="E27">A3</font></i></a>, <a class="txt" href="#E7:39_3"><i><font color="Green" title="E29">A7</font></i></a>, <a class="ref" href="finset_1.html#T26" title="FINSET_1:th.26">FINSET_1:26</a></i>;<br/>
<a NAME="E10:39_3"/>
(  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">N</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> &amp;  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">N</font> )
 
<b>by </b><i><a class="ref" href="waybel_0.html#D6" title="WAYBEL_0:def.6">WAYBEL_0:def 6</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">j</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font><b> such that </b><br/><a NAME="E12:39_3"/>
<font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">N</font>
 <b>and </b><br/><a NAME="E13:39_3"/><i><font color="Green" title="E31">A9</font></i>: 
<font color="Maroon" title="c10">j</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c9">B</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T1" title="WAYBEL_0:th.1">WAYBEL_0:1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_3_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c11">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">G</font> implies <font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">X</font> )</span><br/><b>assume </b><a NAME="E1:39_3_2"/><i><font color="Green" title="E32">A10</font></i>: 
<font color="Maroon" title="c11">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">G</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">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">X</font></span><br/><b>then consider </b><font color="Maroon" title="c12">A</font> being    <a href="yellow19.html#M1" title="YELLOW19:mode.1">Subset</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">N</font>, <font color="Maroon" title="c13">i</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font><b> such that </b><br/><a NAME="E3:39_3_2"/><i><font color="Green" title="E33">A11</font></i>: 
( <font color="Maroon" title="c11">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c12">A</font> &amp; <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i</font> &amp; <font color="Maroon" title="c12">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c13">i</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E8:39_3"><i><font color="Green" title="E30">A8</font></i></a></i>;<br/><a NAME="E4:39_3_2"/>
<font color="Maroon" title="c13">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B</font>
 <b>by </b><i><a class="txt" href="#E7:39_3"><i><font color="Green" title="E29">A7</font></i></a>, <a class="txt" href="#E1:39_3_2"><i><font color="Green" title="E32">A10</font></i></a>, <a class="txt" href="#E3:39_3_2"><i><font color="Green" title="E33">A11</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:39_3_2"/><b>then </b>
<font color="Maroon" title="c13">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c10">j</font>
 <b>by </b><i><a class="txt" href="#E13:39_3"><i><font color="Green" title="E31">A9</font></i></a>, <a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a></i>;<br/><a NAME="E6:39_3_2"/><b>then </b>
<font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c13">i</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_9.html#D7" title="WAYBEL_9:def.7">WAYBEL_9:def 7</a></i>;<br/><a NAME="E7:39_3_2"/><b>then </b>
( <font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c13">i</font></span>)</span> &amp; the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c13">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c2">N</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c13">i</font></span>)</span> )
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="waybel_9.html#D7" title="WAYBEL_9:def.7">WAYBEL_9:def 7</a></i>;<br/><a NAME="E8:39_3_2"/><b>then </b>
( the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c13">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">A</font> &amp; the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c13">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c2">N</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">j</font> )
 <b>by </b><i><a class="txt" href="#E3:39_3_2"><i><font color="Green" title="E33">A11</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E9:39_3_2"/><b>then </b>
( <font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">A</font> &amp; <font color="Maroon" title="c12">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">X</font> )
 <b>by </b><i><a class="txt" href="#E3:39_3_2"><i><font color="Green" title="E33">A11</font></i></a>, <a class="ref" href="pre_topc.html#T48" title="PRE_TOPC:th.48">PRE_TOPC:48</a></i>;<br/><b>hence </b><a NAME="E10:39_3_2"/>
<font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">X</font>
 ; <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="E15:39_3"/>
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c7">G</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> 
 <b>by </b><i><a class="txt" href="#E4:39_3"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 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="E7:39"/><b>then </b><i><font color="Green" title="E27">A12</font></i>: 
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c4">F</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> 
 
<b>by </b><i><a class="txt" href="#E1:39"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="#E5:39"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="compts_1.html#T13" title="COMPTS_1:th.13">COMPTS_1:13</a></i>;<br/>
<a NAME="E8:39"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c4">F</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c5">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c6">x</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">x</font> <a href="waybel_9.html#R3" title="WAYBEL_9:pred.3">is_a_cluster_point_of</a> <font color="Maroon" title="c2">N</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">A</font> be    <a href="yellow19.html#M1" title="YELLOW19:mode.1">Subset</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">N</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c7">A</font></span><br/><a NAME="E1:39_4"/>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font>
 ;<br/><b>hence </b><a NAME="E2:39_4"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c7">A</font>
 <b>by </b><i><a class="txt" href="#E7:39"><i><font color="Green" title="E27">A12</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 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>
<b>hence </b><a NAME="E11:39"/>
<font color="Maroon" title="c6">x</font> <a href="waybel_9.html#R3" title="WAYBEL_9:pred.3">is_a_cluster_point_of</a> <font color="Maroon" title="c2">N</font>
 <b>by </b><i><a class="ref" href="yellow19.html#T31" target="_self" title="YELLOW19:th.31">Th31</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>
