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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="yellow_0.html#V1">lower-bounded</a> <a href="waybel_3.html#V3">continuous</a> <a href="waybel_0.html#NM7">sup-Semilattice</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="waybel23.html#V6" target="_self">with_bottom</a>  <a href="waybel23.html#M1" target="_self">CLbasis</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1"><b>now </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 <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</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> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <span class="p3">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>;<br/><b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="waybel_0.html#NM3">Ideal</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="ref" href="yellow_2.html#T43">YELLOW_2:43</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as  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 <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="yellow_2.html#T7">YELLOW_2:7</a></i>;<br/><a NAME="E3:102_1"/><i><font color="Green">E57</font></i>: 
(  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp;  <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp;  <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp;  <a href="yellow_0.html#R1">ex_sup_of</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> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/><b>thus </b><a NAME="E4:102_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><a href="waybel23.html#K3" target="_self">supMap</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> implies <span class="p1">(<span class="default"><a href="waybel23.html#K5" target="_self">baseMap</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:102_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><a href="waybel23.html#K3" target="_self">supMap</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:102_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="waybel23.html#D10" target="_self">Def10</a></i>;<br/>
<a NAME="E3:102_1_1"/><b>then </b><i><font color="Green">E58</font></i>: 
<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> <span class="p1">(<span class="default"><a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel23.html#E3:102_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="waybel_0.html#T33">WAYBEL_0:33</a></i>;<br/>
<a NAME="E4:102_1_1"/>
 <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">c<sub>6</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></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="E1:102_1_1_1"/><i><font color="Green">E59</font></i>: 
<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>
 ;<br/>

<b>then 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 <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E3:102_1_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="waybel23.html#E1:102_1_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="waybel_3.html#T7">WAYBEL_3:7</a></i>;<br/>
<b>hence </b><a NAME="E4:102_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="waybel23.html#E3:102_1_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="waybel_3.html#T20">WAYBEL_3:20</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:102_1_1"/><b>then </b><i><font color="Green">E59</font></i>: 
<span class="p1">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/>
<a NAME="E6:102_1_1"/>
<span class="p1">(<span class="default"><a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></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="E1:102_1_1_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:102_1_1_2"/><b>then </b><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</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 <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="waybel23.html#E1:102_1_1_2"><i><font color="Green">E60</font></i></a></i>;<br/>
<b>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_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="txt" href="waybel23.html#E2:102_1_1_2"><i><font color="Green">E61</font></i></a>, <a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</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>1</sub></font><b> such that </b><br/><a NAME="E6:102_1_1_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E7:102_1_1_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="waybel23.html#E2:102_1_1_2"><i><font color="Green">E61</font></i></a>, <a class="ref" href="waybel_0.html#D15">WAYBEL_0:def 15</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="txt" href="waybel23.html#E7:102_1_1_2"><i><font color="Green">E63</font></i></a></i>;<br/>
<a NAME="E9:102_1_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="waybel23.html#E6:102_1_1_2"><i><font color="Green">E62</font></i></a>, <a class="ref" href="yellow_0.html#T61">YELLOW_0:61</a></i>;<br/>
<b>hence </b><a NAME="E10:102_1_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="waybel23.html#E7:102_1_1_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="waybel_0.html#D19">WAYBEL_0:def 19</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:102_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="waybel23.html#E5:102_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E8:102_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="waybel23.html#K5" target="_self">baseMap</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</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="waybel23.html#D12" target="_self">Def12</a></i>;<br/>
<b>hence </b><a NAME="E9:102_1_1"/>
<span class="p1">(<span class="default"><a href="waybel23.html#K5" target="_self">baseMap</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E5:102_1"/>
( <span class="p1">(<span class="default"><a href="waybel23.html#K5" target="_self">baseMap</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> implies <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><a href="waybel23.html#K3" target="_self">supMap</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:102_1_2"/>
<span class="p1">(<span class="default"><a href="waybel23.html#K5" target="_self">baseMap</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:102_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="waybel23.html#K5" target="_self">baseMap</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</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="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/>
<a NAME="E3:102_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</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="waybel23.html#D12" target="_self">Def12</a></i>;<br/>
<a NAME="E4:102_1_2"/><b>then </b>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="waybel23.html#E3:102_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="yellow_0.html#T34">YELLOW_0:34</a></i>;<br/>
<a NAME="E5:102_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="waybel23.html#D7" target="_self">Def7</a></i>;<br/>
<b>hence </b><a NAME="E6:102_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><a href="waybel23.html#K3" target="_self">supMap</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="waybel23.html#D10" target="_self">Def10</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:102"/>
<span class="p1"><a href="waybel_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="waybel23.html#K3" target="_self">supMap</a> <span class="p3">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="waybel23.html#K5" target="_self">baseMap</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="waybel_1.html#K1">]</a></span> is <a href="waybel_1.html#V3">Galois</a>
 <b>by </b><i><a class="ref" href="waybel_1.html#T9">WAYBEL_1:9</a></i>;<br/>


</div>
