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

<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="waybel_0.html#V24">up-complete</a> <a href="waybel19.html#NM1">TopPoset</a>;<br/>

<b>assume </b><a NAME="E1:7"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">T</font> is <a href="waybel32.html#V1" target="_self">upper</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon">BB</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Olive">x</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> : verum </span>}</span>  as    <a href="cantor_1.html#M2">prebasis</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/>
<a NAME="E3:7"/>
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> holds <br/> (  <a href="waybel_0.html#K5">downarrow</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">x</font></span><a href="domain_1.html#K6">}</a></span> &amp; ( for <font color="Olive">N</font> being <a href="waybel_0.html#V10">eventually-directed</a> <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font>  st <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="waybel_2.html#K1">sup</a> <font color="Olive">N</font> holds <br/>for <font color="Olive">V</font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Olive">x</font> holds  <font color="Olive">N</font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Olive">V</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>;<br/>

<b>thus </b><a NAME="E1:7_1"/>
 <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:7_1_1"/>
 <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:7_1_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E3:7_1_1_1"/>
for <font color="Olive">G</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">G</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">a'</font> <a href="hidden.html#R2">in</a> <font color="Olive">G</font> holds <br/><span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">G</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>

<b>assume </b><a NAME="E1:7_1_1_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<b>assume </b><a NAME="E2:7_1_1_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">a'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">v</font> = <font color="Maroon">a'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E4:7_1_1_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">v</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#T17">WAYBEL_0:17</a></i>;<br/>
<a NAME="E5:7_1_1_1_1"/>
<font color="Maroon">G</font> is <a href="waybel_0.html#V13">upper</a>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E6:7_1_1_1_1"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 
<b>by </b><i>, , <a class="ref" href="waybel_0.html#D20">WAYBEL_0:def 20</a></i>;<br/>
<a NAME="E7:7_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E8:7_1_1_1_1"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:7_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="pre_topc.html#T54">PRE_TOPC:54</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:7_1_1"/>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:7_1_1_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<b>reconsider </b><font color="Maroon">BB</font> = <font color="Maroon">BB</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E3:7_1_1_2"/><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">BB</font>
 
;<br/>
<a NAME="E4:7_1_1_2"/>
<font color="Maroon">BB</font> is <a href="tops_2.html#V1">open</a>
 
<b>by </b><i><a class="ref" href="yellow_9.html#T28">YELLOW_9:28</a></i>;<br/>
<a NAME="E5:7_1_1_2"/><b>then </b><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a></i>;<br/>
<a NAME="E6:7_1_1_2"/>
<span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font></span>)</span>
 
;<br/>
<a NAME="E7:7_1_1_2"/><b>then </b><i><font color="Green">E56</font></i>: 
 <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#D6">PRE_TOPC:def 6</a></i>;<br/>
<a NAME="E8:7_1_1_2"/>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font>
 
;<br/>
<a NAME="E9:7_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T17">WAYBEL_0:17</a></i>;<br/>
<a NAME="E10:7_1_1_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<b>hence </b><a NAME="E11:7_1_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="ref" href="pre_topc.html#T45">PRE_TOPC:45</a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:7_1"/>
for <font color="Olive">N</font> being <a href="waybel_0.html#V10">eventually-directed</a> <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font>  st <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="waybel_2.html#K1">sup</a> <font color="Olive">N</font> holds <br/>for <font color="Olive">V</font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">x</font> holds  <font color="Olive">N</font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Olive">V</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">N</font> be  <a href="waybel_0.html#V10">eventually-directed</a> <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font>;<br/>

<b>assume </b><a NAME="E1:7_1_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="waybel_2.html#K1">sup</a> <font color="Maroon">N</font>
 ;<br/>

<i><font color="Green">E58</font></i>: <font color="Maroon">x</font> = 
 <a href="yellow_2.html#NK6" target="_self">Sup</a> 
<b>by </b><i>, <a class="ref" href="waybel_2.html#D1">WAYBEL_2:def 1</a></i>
<br/>.= 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K1">netmap</a> <font color="Maroon">N</font>,<font color="Maroon">T</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="yellow_2.html#D5">YELLOW_2:def 5</a></i>
;<br/>
<b>let </b><font color="Maroon">V</font> be    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">x</font>;<br/>

<a NAME="E3:7_1_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a></i>;<br/>
<a NAME="E4:7_1_2"/>
 <a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">BB</font> is    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="yellow_9.html#T23">YELLOW_9:23</a></i>;<br/>
<a NAME="E5:7_1_2"/><b>then </b><i><font color="Green">E60</font></i>: 
the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="cantor_1.html#K1">UniCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">BB</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_9.html#T22">YELLOW_9:22</a></i>;<br/>
<a NAME="E6:7_1_2"/>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T51">TOPS_1:51</a></i>;<br/>
<a NAME="E7:7_1_2"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">Y</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E9:7_1_2"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">BB</font> &amp;  <a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">Y</font> )
 <b>by </b><i>, <a class="ref" href="cantor_1.html#D1">CANTOR_1:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">Y1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:7_1_2"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y1</font> &amp; <font color="Maroon">Y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 <b>by </b><i>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">Z</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E13:7_1_2"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">BB</font> &amp; <font color="Maroon">Z</font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Maroon">Y1</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">Z</font> )
 <b>by </b><i>, , <a class="ref" href="cantor_1.html#D4">CANTOR_1:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">rngN</font> =  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K1">netmap</a> <font color="Maroon">N</font>,<font color="Maroon">T</font></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E15:7_1_2"/>
<font color="Maroon">rngN</font> is <a href="waybel_0.html#V1">directed</a>
 
<b>by </b><i><a class="ref" href="waybel_2.html#T18">WAYBEL_2:18</a></i>;<br/>
<a NAME="E16:7_1_2"/><b>then </b>
 ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">a</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">rngN</font> &amp; ( for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>  st <font color="Olive">b</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">rngN</font> holds <br/><font color="Olive">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b</font> ) )
 
<b>by </b><i><a class="ref" href="waybel_0.html#D39">WAYBEL_0:def 39</a></i>;<br/>
<a NAME="E17:7_1_2"/><b>then </b><i><font color="Green">E67</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">rngN</font>,<font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T15">YELLOW_0:15</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">i</font>, <font color="Olive">j</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>  st <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Olive">j</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">i</font> holds <br/><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E18:7_1_2"/><i><font color="Green">E71</font></i>: 
for <font color="Olive">Q</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> holds <br/> ex <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">Q</font>,<font color="Olive">b</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">Q</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:7_1_2_2"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">Q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/>

<a NAME="E2:7_1_2_2"/><b>then </b>
<font color="Maroon">Q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">BB</font>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">v1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E4:7_1_2_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">v1</font></span>)</span> <a href="subset_1.html#K3">`</a> 
 ;<br/>
<a NAME="E5:7_1_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">v1</font></span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i>, , , , <a class="ref" href="setfam_1.html#T58">SETFAM_1:58</a></i>;<br/>
<a NAME="E6:7_1_2_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">v1</font></span>)</span>
 
;<br/>
<a NAME="E7:7_1_2_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">v1</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E8:7_1_2_2"/><b>then </b><i><font color="Green">E75</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> &amp; not <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">v1</font> )
 
<b>by </b><i><a class="ref" href="waybel_0.html#T17">WAYBEL_0:17</a></i>;<br/>
<a NAME="E9:7_1_2_2"/>
not <font color="Maroon">rngN</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">v1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_2_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:7_1_2_2_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">rngN</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">v1</font>
 ;<br/>

<a NAME="E2:7_1_2_2_1"/>
 <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">v1</font>,<font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T34">WAYBEL_0:34</a></i>;<br/>
<a NAME="E3:7_1_2_2_1"/><b>then </b>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">rngN</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">v1</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="yellow_0.html#T34">YELLOW_0:34</a></i>;<br/>
<b>hence </b><a NAME="E4:7_1_2_2_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="waybel_0.html#T34">WAYBEL_0:34</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">w</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:7_1_2_2"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">rngN</font> &amp; not <font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">v1</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">w'</font> = <font color="Maroon">w</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/>
<a NAME="E13:7_1_2_2"/>
<font color="Maroon">w'</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">v1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E14:7_1_2_2"/><b>then </b><i><font color="Green">E80</font></i>: 
<font color="Maroon">w'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">i</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E16:7_1_2_2"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font>
 <b>and </b><br/><a NAME="E17:7_1_2_2"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">w'</font> <a href="hidden.html#R1">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> <b>by </b><i/>;<br/>
<a NAME="E19:7_1_2_2"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font> &amp; <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> )
 
<b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E21:7_1_2_2"/><i><font color="Green">E84</font></i>: 
for <font color="Olive">l</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>  st <font color="Maroon">b</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">l</font> holds <br/><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">l</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T11">WAYBEL_0:11</a></i>;<br/>
<b>take </b>
<font color="Maroon">b</font>
;<br/>

<b>thus </b><a NAME="E22:7_1_2_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font>
 ;<br/>

<b>thus </b><a NAME="E23:7_1_2_2"/>
for <font color="Olive">j</font>, <font color="Olive">k</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>  st <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Olive">k</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">j</font> holds <br/><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_2_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>, <font color="Maroon">k</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>;<br/>

<b>assume </b><a NAME="E1:7_1_2_2_2"/><i><font color="Green">E87</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">k</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> )
 ;<br/>

<a NAME="E2:7_1_2_2_2"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">k</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:7_1_2_2_2"/>
<font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">v1</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:7_1_2_2_2"/><b>then </b>
( <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> &amp; not <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">v1</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E5:7_1_2_2_2"/><b>then </b>
( <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> &amp; not <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">v1</font> )
 
<b>by </b><i><a class="ref" href="waybel_0.html#T17">WAYBEL_0:17</a></i>;<br/>
<a NAME="E6:7_1_2_2_2"/><b>then </b>
not <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">k</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">v1</font>
 
<b>by </b><i>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/>
<a NAME="E7:7_1_2_2_2"/><b>then </b>
not <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">v1</font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T17">WAYBEL_0:17</a></i>;<br/>
<a NAME="E8:7_1_2_2_2"/><b>then </b>
<font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">v1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E9:7_1_2_2_2"/>
<font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E20:7_1_2"/><i><font color="Green">E90</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Z</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font> )
 <b>and </b><br/><a NAME="E21:7_1_2"/><i><font color="Green">E91</font></i>: 
for <font color="Olive">Q</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">Q</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">Q</font>]
 <b>from </b><i><a class="ref" href="wellord2.html#S1">WELLORD2:sch 1</a>();<br/></i>
<b>reconsider </b><font color="Maroon">rngf</font> =  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> as  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">N</font></span>)</span> <b>by </b><i>, , <a class="ref" href="finset_1.html#T26">FINSET_1:26</a></i>;<br/>
<a NAME="E23:7_1_2"/>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">N</font> is <a href="waybel_0.html#V1">directed</a>
 
<b>by </b><i><a class="ref" href="waybel_0.html#D6">WAYBEL_0:def 6</a></i>;<br/>
<b>then consider </b><font color="Maroon">k</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E25:7_1_2"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">N</font> &amp; <font color="Maroon">k</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">rngf</font> )
 <b>by </b><i><a class="ref" href="waybel_0.html#T1">WAYBEL_0:1</a></i>;<br/>
<b>take </b>
<font color="Maroon">k</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D11">WAYBEL_0:def 11</a><br/>

<b>let </b><font color="Maroon">k1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>;<br/>

<b>assume </b><a NAME="E26:7_1_2"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">k</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">k1</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Q</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:7_1_2_3"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">Q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/><a NAME="E2:7_1_2_3"/><b>then </b><i><font color="Green">E97</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">Q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">rngf</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon">j</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">Q</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> ;<br/><a NAME="E4:7_1_2_3"/>
<font color="Maroon">j</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i>, , <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/><a NAME="E5:7_1_2_3"/><b>then </b>
<font color="Maroon">j</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">k1</font>
 <b>by </b><i>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/><b>hence </b><a NAME="E6:7_1_2_3"/>
<font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">k1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<a NAME="E28:7_1_2"/><b>then </b>
( <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">k1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y1</font> &amp; <font color="Maroon">Y1</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">Y</font> )
 
<b>by </b><i>, , <a class="ref" href="setfam_1.html#T58">SETFAM_1:58</a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E29:7_1_2"/><b>then </b>
( <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">k1</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font> &amp;  <a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> )
 
<b>by </b><i>, <a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>;<br/>
<b>hence </b><a NAME="E30:7_1_2"/>
<font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">k1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 ;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:7"/>
<font color="Maroon">T</font> is <a href="waybel32.html#V2" target="_self">order_consistent</a>
 <b>by </b><i/>;<br/>


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