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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel11.html#V4">Scott</a> <a href="waybel_9.html#NM1">TopLattice</a>;<br/>

<b>assume </b><a NAME="E1:42"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="waybel_3.html#V3">continuous</a>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E2:42"/><i><font color="Green">E29</font></i>: 
 <a href="waybel11.html#K5">sigma</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="waybel14.html#T23" target="_self">Th23</a></i>;<br/>
<a NAME="E3:42"/><i><font color="Green">E30</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/>
<a NAME="E4:42"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> holds <br/>( not  <a href="waybel_3.html#K1">waybelow</a> <font color="Olive">b<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="waybel_3.html#K1">waybelow</a> <font color="Olive">b<sub>1</sub></font> is <a href="waybel_0.html#V1">directed</a> )
 
;<br/>
<a NAME="E5:42"/>
 <a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> is <a href="waybel_3.html#V2">satisfying_axiom_of_approximation</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_3.html#D5">WAYBEL_3:def 5</a><br/>

<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_3.html#K2">wayabove</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span> ;<br/>
<a NAME="E1:42_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E2:42"><i><font color="Green">E29</font></i></a>, <a class="txt" href="waybel14.html#E3:42"><i><font color="Green">E30</font></i></a></i>;<br/>
<a NAME="E2:42_1"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_3.html#K2">wayabove</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="waybel14.html#E1:42"><i><font color="Green">E28</font></i></a>, <a class="ref" href="waybel14.html#T33" target="_self">Th33</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:42_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:42_1_1_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_3.html#K2">wayabove</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span>  )
 <b>by </b><i><a class="txt" href="waybel14.html#E2:42_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:42_1_1_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="waybel14.html#E3:42_1_1_1"><i><font color="Green">E33</font></i></a></i>;<br/><a NAME="E6:42_1_1_1"/>
 <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">c<sub>8</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="waybel14.html#E1:42"><i><font color="Green">E28</font></i></a>, <a class="ref" href="waybel11.html#T36">WAYBEL11:36</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="waybel14.html#E3:42"><i><font color="Green">E30</font></i></a>, <a class="txt" href="waybel14.html#E5:42_1_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/><a NAME="E8:42_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_3.html#D1">WAYBEL_3:def 1</a><br/>

<b>assume </b><a NAME="E1:42_1_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<a NAME="E2:42_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/>
<a NAME="E3:42_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T22">YELLOW_1:22</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:42_1_1_1_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="waybel14.html#E5:42_1_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E6:42_1_1_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E2:42"><i><font color="Green">E29</font></i></a>, <a class="txt" href="waybel14.html#E3:42"><i><font color="Green">E30</font></i></a>, <a class="txt" href="waybel14.html#E5:42_1_1_1_1"><i><font color="Green">E35</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E8:42_1_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="waybel14.html#E3:42"><i><font color="Green">E30</font></i></a>, <a class="txt" href="waybel14.html#E5:42_1_1_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E9:42_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> is <a href="waybel_0.html#V13">upper</a>
 
<b>by </b><i><a class="ref" href="waybel11.html#D4">WAYBEL11:def 4</a></i>;<br/>
<a NAME="E10:42_1_1_1_1"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E5:42_1_1_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="waybel11.html#T42">WAYBEL11:42</a></i>;<br/>
<a NAME="E11:42_1_1_1_1"/>
 <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="waybel_3.html#T11">WAYBEL_3:11</a></i>;<br/>
<a NAME="E12:42_1_1_1_1"/><b>then </b><i><font color="Green">E37</font></i>: 
 <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E10:42_1_1_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="waybel14.html#E5:42_1_1_1_1"><i><font color="Green">E35</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>13</sub></font>
;<br/>

<b>thus </b><a NAME="E14:42_1_1_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="waybel14.html#E5:42_1_1_1_1"><i><font color="Green">E35</font></i></a></i>;<br/>

<b>thus </b><a NAME="E15:42_1_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="waybel14.html#E5:42_1_1_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="waybel14.html#E12:42_1_1_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E9:42_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>hence </b><a NAME="E10:42_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel14.html#E3:42_1_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:42_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:42_1_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="waybel14.html#E4:42_1_1"><i><font color="Green">E33</font></i></a></i>;<br/><a NAME="E6:42_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="waybel14.html#E4:42_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="waybel_3.html#T7">WAYBEL_3:7</a></i>;<br/><a NAME="E7:42_1_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="waybel_3.html#T1">WAYBEL_3:1</a></i>;<br/><a NAME="E8:42_1_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><b>hence </b><a NAME="E9:42_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="waybel14.html#E4:42_1_1"><i><font color="Green">E33</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:42_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<b>hence </b><a NAME="E5:42_1"/>
<font color="Maroon">c<sub>3</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="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_1.html#T22">YELLOW_1:22</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:42"/>
 <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> is <a href="waybel_3.html#V3">continuous</a>
 <b>by </b><i><a class="txt" href="waybel14.html#E2:42"><i><font color="Green">E29</font></i></a>, <a class="txt" href="waybel14.html#E4:42"><i><font color="Green">E31</font></i></a>, <a class="ref" href="waybel_3.html#D6">WAYBEL_3:def 6</a></i>;<br/>


</div>
