<?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_0.html#NM7">sup-Semilattice</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <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> <span class="p2">(<span class="default"><a href="waybel_4.html#K3" target="_self">Aux</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="orders_3.html#D5">ORDERS_3:def 5</a><br/>



<a NAME="E1:66"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</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_4.html#K3" target="_self">Aux</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</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_4.html#K3" target="_self">Aux</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> )
 
;<br/>
<a NAME="E2:66"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_4.html#K3" target="_self">Aux</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_4.html#K3" target="_self">Aux</a> <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/>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> as  <a href="waybel_4.html#V5" target="_self">auxiliary</a> <a href="alg_1.html#NM1">Relation</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="waybel_4.html#D9" target="_self">Def9</a></i>;<br/>
<b>assume </b><a NAME="E4:66"/>
<font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E5:66"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>2</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>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel_4.html#K8" target="_self">MonSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>

<b>assume </b><a NAME="E6:66"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K9" target="_self">Rel2Map</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K9" target="_self">Rel2Map</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<b>thus </b><a NAME="E7:66"/>
<font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><b>proof </b></a><div class="add">

<a NAME="E1:66_1"/><i><font color="Green">E41</font></i>: 
( <span class="p1">(<span class="default"><a href="waybel_4.html#K9" target="_self">Rel2Map</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="waybel_4.html#K7" target="_self">-below</a>  &amp; <span class="p1">(<span class="default"><a href="waybel_4.html#K9" target="_self">Rel2Map</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="waybel_4.html#K7" target="_self">-below</a>  )
 
<b>by </b><i><a class="ref" href="waybel_4.html#D15" target="_self">Def15</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<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> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:66_1"/><i><font color="Green">E42</font></i>: 
( <span class="p1">(<span class="default"><a href="waybel_4.html#K9" target="_self">Rel2Map</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> is <a href="orders_3.html#V5">monotone</a> &amp; ( 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 <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="ref" href="waybel_4.html#D14" target="_self">Def14</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<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> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E5:66_1"/><i><font color="Green">E43</font></i>: 
( <span class="p1">(<span class="default"><a href="waybel_4.html#K9" target="_self">Rel2Map</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> is <a href="orders_3.html#V5">monotone</a> &amp; ( 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 <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="ref" href="waybel_4.html#D14" target="_self">Def14</a></i>;<br/>
<a NAME="E6:66_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="yellow_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_2.html#D1">YELLOW_2:def 1</a><br/>

<b>assume </b><a NAME="E1:66_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then 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 <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E3:66_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_4.html#E1:66_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="waybel_4.html#E3:66_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="waybel_4.html#D13" target="_self">Def13</a></i>;<br/>
<a NAME="E4:66_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_4.html#E1:66_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="waybel_4.html#E5:66_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="waybel_4.html#D13" target="_self">Def13</a></i>;<br/>
<a NAME="E5:66_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_4.html#E5:66"><i><font color="Green">E39</font></i></a>, <a class="ref" href="waybel_4.html#T29" target="_self">Th29</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</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> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> ;<br/>
<a NAME="E7:66_1_1"/>
<font color="Maroon">c<sub>12</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="waybel_4.html#E3:66_1_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="waybel_4.html#E4:66_1_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="waybel_4.html#E5:66_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/>
<b>hence </b><a NAME="E8:66_1_1"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</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> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E7:66_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="waybel_4.html#K7" target="_self">-below</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="waybel_4.html#K7" target="_self">-below</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="waybel_4.html#K8" target="_self">MonSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_4.html#E1:66_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="waybel_4.html#E3:66_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="waybel_4.html#E5:66_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="waybel_4.html#D14" target="_self">Def14</a></i>;<br/>
<b>hence </b><a NAME="E8:66_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="waybel_4.html#E6:66"><i><font color="Green">E40</font></i></a>, <a class="txt" href="waybel_4.html#E1:66_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>


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


</div>
