<?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="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>;<br/>



<b>assume </b><a NAME="E1:64"/><i><font color="Green">E37</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>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</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="E2:64"/>
<font color="Maroon">c<sub>4</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>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E4:64"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">c<sub>5</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</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="E1:64_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E2:64_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</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="E4:64_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>
<a NAME="E5:64_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span> <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="waybel_4.html#E1:64"><i><font color="Green">E37</font></i></a>, <a class="txt" href="waybel_4.html#E4:64_1"><i><font color="Green">E39</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:64_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="waybel_4.html#E4:64_1"><i><font color="Green">E39</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_4.html#K7" target="_self">-below</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>7</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="waybel_4.html#K7" target="_self">-below</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</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/>
<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

<a NAME="E6:64"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_4.html#K7" target="_self">-below</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">c<sub>5</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="waybel_4.html#K7" target="_self">-below</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="ref" href="waybel_4.html#D13" target="_self">Def13</a></i>;<br/>
<b>hence </b><a NAME="E7:64"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_4.html#K7" target="_self">-below</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="waybel_4.html#K7" target="_self">-below</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> &amp; <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#E4:64"><i><font color="Green">E38</font></i></a>, <a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/>


</div>
