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

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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><br/> for <font color="Olive" title="b2">s</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> holds <br/> ( <font color="Olive" title="b2">s</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> iff for <font color="Olive" title="b3">A</font> being   <a href="yellow19.html#M1" title="YELLOW19:mode.1">Subset</a> of <font color="Maroon" title="c1">T</font>,<font color="Olive" title="b1">N</font> holds  <font color="Olive" title="b2">s</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="Olive" title="b3">A</font> )</span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">s</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> holds <br/> ( <font color="Olive" title="b1">s</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> iff for <font color="Olive" title="b2">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> holds  <font color="Olive" title="b1">s</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="Olive" title="b2">A</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">s</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</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="c3">s</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> iff for <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> holds  <font color="Maroon" title="c3">s</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="Olive" title="b1">A</font> )</span><br/>





<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <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> holds  <font color="Maroon" title="c3">s</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="Olive" title="b1">A</font> ) implies <font color="Maroon" title="c3">s</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>
<div class="add"><span class="kw">assume </span><a NAME="E1:39_1"/><span class="lab"><font color="Green" title="E24">A1</font></span>: 
<font color="Maroon" title="c3">s</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <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> holds  <font color="Maroon" title="c3">s</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="Olive" title="b1">A</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">s</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="c4">A</font></span><br/><a NAME="E2:39_1"/>
<font color="Maroon" title="c2">N</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow19.html#T9" target="_self" title="YELLOW19:th.9">Th9</a></span>;<br/><span class="kw">hence </span><a NAME="E3:39_1"/>
<font color="Maroon" title="c3">s</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="c4">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="ref" href="yellow19.html#T23" target="_self" title="YELLOW19:th.23">Th23</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>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:39"/><span class="lab"><font color="Green" title="E24">A2</font></span>: 
for <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> holds  <font color="Maroon" title="c3">s</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="Olive" title="b1">A</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="c3">s</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/>

<span class="kw">let </span><font color="Maroon" title="c4">V</font> be    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c3">s</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="waybel_9.html#D9" title="WAYBEL_9:def.9">WAYBEL_9:def 9</a> <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="c2">N</font> <a href="waybel_0.html#R2" title="WAYBEL_0:pred.2">is_often_in</a> <font color="Maroon" title="c4">V</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">i</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="waybel_0.html#D12" title="WAYBEL_0:def.12">WAYBEL_0:def 12</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">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">N</font> st <br/>( <font color="Maroon" title="c5">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">V</font> )</span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c6">A</font> =  <a href="relset_1.html#K2" title="RELSET_1:func.2">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="c5">i</font></span>)</span> as    <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> <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow19.html#D2" target="_self" title="YELLOW19:def.2">Def2</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c4">V</font></span>)</span>;<br/>
<a NAME="E5:39"/><span class="lab"><font color="Green" title="E25">A3</font></span>: 
<font color="Maroon" title="c3">s</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="c4">V</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="E6:39"/>
<font color="Maroon" title="c3">s</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="c6">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39"><span class="lab"><font color="Green" title="E24">A2</font></span></a></span>;<br/>
<a NAME="E7:39"/><span class="kw">then </span>
<font color="Maroon" title="c6">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c4">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:39"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="ref" href="pre_topc.html#D13" title="PRE_TOPC:def.13">PRE_TOPC:def 13</a></span>;<br/>
<a NAME="E8:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A4</font></span>: 
<font color="Maroon" title="c6">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c4">V</font></span>)</span> <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">T</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="E9:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A5</font></span>: 
<font color="Maroon" title="c7">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="c4">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E10:39"/><span class="lab"><font color="Green" title="E28">A6</font></span>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c4">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">V</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="E11:39"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:39"><span class="lab"><font color="Green" title="E26">A4</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">j</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E13:39"/><span class="lab"><font color="Green" title="E29">A7</font></span>: 
<font color="Maroon" title="c8">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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="c5">i</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E14:39"/><span class="lab"><font color="Green" title="E30">A8</font></span>: 
<font color="Maroon" title="c7">x</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 <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="c5">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E15:39"/><span class="lab"><font color="Green" title="E31">A9</font></span>: 
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="c5">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">l</font> where <font color="Olive" title="b1">l</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font> : <font color="Maroon" title="c5">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">l</font> </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_9.html#T12" title="WAYBEL_9:th.12">WAYBEL_9:12</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">j</font> = <font color="Maroon" title="c8">j</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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="c5">i</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:39"><span class="lab"><font color="Green" title="E29">A7</font></span></a></span>;<br/>
<a NAME="E17:39"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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="c5">i</font></span>)</span> <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 <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="c5">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">l</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font><span class="kw"> such that </span><br/><a NAME="E19:39"/><span class="lab"><font color="Green" title="E32">A10</font></span>: 
<font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">l</font>
 <span class="kw">and </span><br/><a NAME="E20:39"/><span class="lab"><font color="Green" title="E33">A11</font></span>: 
<font color="Maroon" title="c5">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c10">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:39"><span class="lab"><font color="Green" title="E29">A7</font></span></a>, <a class="txt" href="#E15:39"><span class="lab"><font color="Green" title="E31">A9</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c10">l</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="c5">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c10">l</font> &amp; <font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c10">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">V</font> )</span><br/>

<span class="kw">thus </span><a NAME="E21:39"/>
<font color="Maroon" title="c5">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c10">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:39"><span class="lab"><font color="Green" title="E33">A11</font></span></a></span>; <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="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c10">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">V</font></span><br/>

<font color="Maroon" title="c7">x</font> = 
<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="c5">i</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c9">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:39"><span class="lab"><font color="Green" title="E30">A8</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c10">l</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:39"><span class="lab"><font color="Green" title="E32">A10</font></span></a>, <a class="ref" href="waybel_9.html#T16" title="WAYBEL_9:th.16">WAYBEL_9:16</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E23:39"/>
<font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c10">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:39"><span class="lab"><font color="Green" title="E27">A5</font></span></a>, <a class="txt" href="#E10:39"><span class="lab"><font color="Green" title="E28">A6</font></span></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/>


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