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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="waybel_0.html#V24">up-complete</a> <a href="waybel11.html#V4">Scott</a> <a href="waybel19.html#NM1">TopPoset</a>;<br/>

<a NAME="E1:13"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>(  <a href="waybel_0.html#K5">downarrow</a> <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span> &amp; ( for <font color="Olive">b<sub>2</sub></font> being <a href="waybel_0.html#V10">eventually-directed</a> <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel_2.html#K1">sup</a> <font color="Olive">b<sub>2</sub></font> holds <br/>for <font color="Olive">b<sub>3</sub></font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Olive">b<sub>1</sub></font> holds <font color="Olive">b<sub>2</sub></font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Olive">b<sub>3</sub></font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E1:13_1"/>
for <font color="Olive">b<sub>1</sub></font> being <a href="waybel_0.html#V10">eventually-directed</a> <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel_2.html#K1">sup</a> <font color="Olive">b<sub>1</sub></font> holds <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">c<sub>2</sub></font> holds <font color="Olive">b<sub>1</sub></font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Olive">b<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="waybel_0.html#V10">eventually-directed</a> <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:13_1_1"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel_2.html#K1">sup</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:13_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#NK6" target="_self">Sup</a> 
 
<b>by </b><i><a class="txt" href="waybel32.html#E1:13_1_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="waybel_2.html#D1">WAYBEL_2:def 1</a></i>;<br/>
<a NAME="E3:13_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_2.html#D5">YELLOW_2:def 5</a></i>;<br/>
<a NAME="E4:13_1_1"/><b>then </b><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <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">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
;<br/>
<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<a NAME="E5:13_1_1"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a></i>;<br/>
<a NAME="E6:13_1_1"/><i><font color="Green">E11</font></i>: 
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>4</sub></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="E7:13_1_1"/><b>then </b><i><font color="Green">E12</font></i>: 
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>4</sub></font> is <a href="waybel11.html#V1">inaccessible_by_directed_joins</a>
 
<b>by </b><i><a class="ref" href="waybel11.html#D4">WAYBEL11:def 4</a></i>;<br/>
<a NAME="E8:13_1_1"/><i><font color="Green">E13</font></i>: 
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>4</sub></font> is  <a href="waybel_0.html#V13">upper</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="waybel32.html#E6:13_1_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="waybel11.html#D4">WAYBEL11:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E10:13_1_1"/>
<font color="Maroon">c<sub>5</sub></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="E11:13_1_1"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="waybel32.html#E4:13_1_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="waybel32.html#E5:13_1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="waybel32.html#E7:13_1_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="waybel11.html#D1">WAYBEL11:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E13:13_1_1"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="waybel32.html#E13:13_1_1"><i><font color="Green">E14</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E16:13_1_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>8</sub></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">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E17:13_1_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="waybel32.html#E13:13_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="waybel32.html#E16:13_1_1"><i><font color="Green">E15</font></i></a></i>;<br/>
<a NAME="E19:13_1_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="waybel32.html#E17:13_1_1"><i><font color="Green">E16</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E21:13_1_1"/><i><font color="Green">E18</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font> st <font color="Maroon">c<sub>10</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Olive">b<sub>1</sub></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">c<sub>10</sub></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">c<sub>11</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>

<b>assume </b><a NAME="E22:13_1_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/>

<a NAME="E23:13_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="waybel32.html#E21:13_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="waybel32.html#E22:13_1_1"><i><font color="Green">E19</font></i></a></i>;<br/>
<a NAME="E24:13_1_1"/><b>then </b><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="waybel32.html#E8:13_1_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="waybel32.html#E13:13_1_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="waybel32.html#E19:13_1_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="waybel_0.html#D20">WAYBEL_0:def 20</a></i>;<br/>
<a NAME="E25:13_1_1"/>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>;<br/>
<b>hence </b><a NAME="E26:13_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="waybel32.html#E24:13_1_1"><i><font color="Green">E20</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:13_1"/>
(  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> &amp; ( for <font color="Olive">b<sub>1</sub></font> being <a href="waybel_0.html#V10">eventually-directed</a> <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel_2.html#K1">sup</a> <font color="Olive">b<sub>1</sub></font> holds <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">c<sub>2</sub></font> holds <font color="Olive">b<sub>1</sub></font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Olive">b<sub>2</sub></font> ) )
 <b>by </b><i><a class="ref" href="waybel32.html#T9" target="_self">Th9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:13"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="waybel32.html#V2" target="_self">order_consistent</a>
 <b>by </b><i><a class="ref" href="waybel32.html#D2" target="_self">Def2</a></i>;<br/>


</div>
