<?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="waybel_0.html#NM8">LATTICE</a>;<br/>

<b>assume </b><a NAME="E1:53"/><i><font color="Green">E34</font></i>: 
 <a href="waybel_6.html#K4" target="_self">PRIME</a> <font color="Maroon">c<sub>1</sub></font> is <a href="waybel_6.html#V4" target="_self">order-generating</a>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="funct_3.html#K5">chi</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<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> is <a href="waybel_6.html#V5" target="_self">prime</a> </span>}</span> ;<br/>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being  <a href="waybel_6.html#V5" target="_self">prime</a> <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E3:53"/><i><font color="Green">E35</font></i>: 
 <a href="funct_3.html#K5">chi</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="funct_3.html#K5">chi</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<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> is <a href="waybel_6.html#V5" target="_self">prime</a> </span>}</span> 
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:53_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="funct_3.html#K5">chi</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<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> is <a href="waybel_6.html#V5" target="_self">prime</a> </span>}</span> 
 ;<br/><a NAME="E2:53_1"/><b>then </b>
ex <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> st <br/>( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="funct_3.html#K5">chi</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="waybel_6.html#V5" target="_self">prime</a> )
 ;<br/><b>hence </b><a NAME="E3:53_1"/>
<font color="Maroon">c<sub>4</sub></font> is   <a href="funct_1.html#NM1">Function</a>
 ;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="funct_3.html#K5">chi</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<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> is <a href="waybel_6.html#V5" target="_self">prime</a> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="fraenkel.html#V1">functional</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="waybel_6.html#E3:53"><i><font color="Green">E35</font></i></a>, <a class="ref" href="fraenkel.html#D1">FRAENKEL:def 1</a></i>;<br/>
<i><font color="Green">E36</font></i>: the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
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="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="yellow_1.html#T4">YELLOW_1:4</a></i>
<br/>.= 
 <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>
;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 </span>}</span> ;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E8:53"/><i><font color="Green">E37</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E9:53"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</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> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="funct_1.html#S3">FUNCT_1:sch 3</a>();<br/></i>
<a NAME="E10:53"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_3"><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="E1:53_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</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="E3:53_3"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:53_3"/><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"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 1 </span>}</span> 
 
<b>by </b><i><a class="txt" href="waybel_6.html#E8:53"><i><font color="Green">E37</font></i></a>, <a class="txt" href="waybel_6.html#E9:53"><i><font color="Green">E38</font></i></a>, <a class="txt" href="waybel_6.html#E3:53_3"><i><font color="Green">E39</font></i></a></i>;<br/>
<a NAME="E5:53_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</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="E1:53_3_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<a NAME="E2:53_3_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> st <br/>( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="txt" href="waybel_6.html#E4:53_3"><i><font color="Green">E40</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:53_3_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:53_3"/>
<font color="Maroon">c<sub>6</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#K3">BoolePoset</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_6.html#E6:53"><i><font color="Green">E36</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <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#K3">BoolePoset</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <b>by </b><i><a class="txt" href="waybel_6.html#E8:53"><i><font color="Green">E37</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<a NAME="E12:53"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="waybel_0.html#V19">meet-preserving</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D34">WAYBEL_0:def 34</a><br/>

<b>assume </b><a NAME="E1:53_4"/>
 <a href="yellow_0.html#R2">ex_inf_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K7">}</a></span>,<font color="Maroon">c<sub>1</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D30">WAYBEL_0:def 30</a><br/>

<b>thus </b><a NAME="E2:53_4"/>
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">c<sub>6</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K7">}</a></span>, <a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>

<a NAME="E3:53_4"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i><a class="txt" href="waybel_6.html#E8:53"><i><font color="Green">E37</font></i></a>, <a class="ref" href="funct_1.html#T118">FUNCT_1:118</a></i>;<br/>
<a NAME="E4:53_4"/><i><font color="Green">E41</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 1 </span>}</span>  <a href="xboole_0.html#K3">/\</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 1 </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1 </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</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="E1:53_4_1"/>
<font color="Maroon">c<sub>9</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="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 1 </span>}</span>  <a href="xboole_0.html#K3">/\</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 1 </span>}</span> 
 ;<br/>

<a NAME="E2:53_4_1"/><b>then </b><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>9</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="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 1 </span>}</span>  &amp; <font color="Maroon">c<sub>9</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="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 1 </span>}</span>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E4:53_4_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 1 )
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E6:53_4_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="txt" href="waybel_6.html#E2:53_4_1"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E7:53_4_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>12</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="E9:53_4_1"/><i><font color="Green">E45</font></i>: 
(  <a href="funct_3.html#K5">chi</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> is <a href="waybel_6.html#V5" target="_self">prime</a> )
 ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <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#K3">BoolePoset</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <b>by </b><i><a class="txt" href="waybel_6.html#E9:53_4_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="waybel_6.html#T31" target="_self">Th31</a></i>;<br/>
<a NAME="E11:53_4_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E12:53_4_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T118">FUNCT_1:118</a></i>;<br/>
<a NAME="E13:53_4_1"/>
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/>( <font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  iff <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> )
 
<b>by </b><i><a class="txt" href="waybel_6.html#E9:53_4_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="waybel_6.html#T32" target="_self">Th32</a></i>;<br/>
<a NAME="E14:53_4_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> is <a href="waybel_0.html#V19">meet-preserving</a>
 
<b>by </b><i><a class="txt" href="waybel_6.html#E9:53_4_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="waybel_6.html#T25" target="_self">Th25</a></i>;<br/>
<a NAME="E15:53_4_1"/><b>then </b>
( <font color="Maroon">c<sub>13</sub></font> <a href="waybel_0.html#R3">preserves_inf_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K7">}</a></span> &amp;  <a href="yellow_0.html#R2">ex_inf_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K7">}</a></span>,<font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K7">}</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="domain_1.html#K7">}</a></span> )
 
<b>by </b><i><a class="ref" href="waybel_0.html#D34">WAYBEL_0:def 34</a>, <a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a>, <a class="ref" href="yellow_0.html#T40">YELLOW_0:40</a></i>;<br/>
<b>then </b><font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="waybel_6.html#E12:53_4_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="waybel_0.html#D30">WAYBEL_0:def 30</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="waybel_6.html#E4:53_4_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="waybel_6.html#E6:53_4_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="yellow_5.html#T2">YELLOW_5:2</a></i>
;<br/>
<b>hence </b><a NAME="E17:53_4_1"/>
<font color="Maroon">c<sub>9</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="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1 </span>}</span> 
 <b>by </b><i><a class="txt" href="waybel_6.html#E4:53_4_1"><i><font color="Green">E43</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:53_4"/><i><font color="Green">E42</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1 </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 1 </span>}</span>  <a href="xboole_0.html#K3">/\</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 1 </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</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="E1:53_4_2"/>
<font color="Maroon">c<sub>9</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="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1 </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E3:53_4_2"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1 )
 ;<br/>
<a NAME="E4:53_4_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>11</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="E6:53_4_2"/><i><font color="Green">E44</font></i>: 
(  <a href="funct_3.html#K5">chi</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> is <a href="waybel_6.html#V5" target="_self">prime</a> )
 ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <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#K3">BoolePoset</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <b>by </b><i><a class="txt" href="waybel_6.html#E6:53_4_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="waybel_6.html#T31" target="_self">Th31</a></i>;<br/>
<a NAME="E8:53_4_2"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E9:53_4_2"/><b>then </b><i><font color="Green">E45</font></i>: 
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T118">FUNCT_1:118</a></i>;<br/>
<a NAME="E10:53_4_2"/><i><font color="Green">E46</font></i>: 
1 <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K4">Top</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="yellow_1.html#T19">YELLOW_1:19</a></i>;<br/>
<a NAME="E11:53_4_2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K4">Top</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K4">Top</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T45">YELLOW_0:45</a></i>;<br/>
<a NAME="E12:53_4_2"/>
<font color="Maroon">c<sub>12</sub></font> is <a href="waybel_0.html#V19">meet-preserving</a>
 
<b>by </b><i><a class="txt" href="waybel_6.html#E6:53_4_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="waybel_6.html#T33" target="_self">Th33</a></i>;<br/>
<a NAME="E13:53_4_2"/><b>then </b>
( <font color="Maroon">c<sub>12</sub></font> <a href="waybel_0.html#R3">preserves_inf_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K7">}</a></span> &amp;  <a href="yellow_0.html#R2">ex_inf_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K7">}</a></span>,<font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K7">}</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="domain_1.html#K7">}</a></span> )
 
<b>by </b><i><a class="ref" href="waybel_0.html#D34">WAYBEL_0:def 34</a>, <a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a>, <a class="ref" href="yellow_0.html#T40">YELLOW_0:40</a></i>;<br/>
<a NAME="E14:53_4_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_6.html#E9:53_4_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="waybel_0.html#D30">WAYBEL_0:def 30</a></i>;<br/>
<a NAME="E15:53_4_2"/><b>then </b><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_0.html#K4">Top</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_0.html#K4">Top</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> )
 
<b>by </b><i><a class="txt" href="waybel_6.html#E3:53_4_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="waybel_6.html#E10:53_4_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/>
<a NAME="E16:53_4_2"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="waybel_6.html#E10:53_4_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="waybel_6.html#E11:53_4_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="orders_2.html#T25">ORDERS_2:25</a></i>;<br/>
<a NAME="E17:53_4_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="waybel_6.html#E10:53_4_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="waybel_6.html#E11:53_4_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="waybel_6.html#E15:53_4_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="orders_2.html#T25">ORDERS_2:25</a></i>;<br/>
<a NAME="E18:53_4_2"/><b>then </b>
( <font color="Maroon">c<sub>9</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="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 1 </span>}</span>  &amp; <font color="Maroon">c<sub>9</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="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 1 </span>}</span>  )
 
<b>by </b><i><a class="txt" href="waybel_6.html#E3:53_4_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="waybel_6.html#E16:53_4_2"><i><font color="Green">E49</font></i></a></i>;<br/>
<b>hence </b><a NAME="E19:53_4_2"/>
<font color="Maroon">c<sub>9</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="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 1 </span>}</span>  <a href="xboole_0.html#K3">/\</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 1 </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b> <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K7">}</a></span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="waybel_6.html#E3:53_4"><i><font color="Green">E40</font></i></a>, <a class="ref" href="yellow_0.html#T40">YELLOW_0:40</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_1.html#T17">YELLOW_1:17</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 1 </span>}</span>  <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="waybel_6.html#E9:53"><i><font color="Green">E38</font></i></a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 1 </span>}</span>  <a href="xboole_0.html#K3">/\</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 1 </span>}</span> 
<b>by </b><i><a class="txt" href="waybel_6.html#E9:53"><i><font color="Green">E38</font></i></a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1 </span>}</span> 
<b>by </b><i><a class="txt" href="waybel_6.html#E4:53_4"><i><font color="Green">E41</font></i></a>, <a class="txt" href="waybel_6.html#E5:53_4"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="waybel_6.html#E9:53"><i><font color="Green">E38</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K7">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="yellow_0.html#T40">YELLOW_0:40</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E13:53"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="waybel_0.html#V20">join-preserving</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D35">WAYBEL_0:def 35</a><br/>

<b>assume </b><a NAME="E1:53_5"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K7">}</a></span>,<font color="Maroon">c<sub>1</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a><br/>

<b>thus </b><a NAME="E2:53_5"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>6</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K7">}</a></span>, <a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>

<a NAME="E3:53_5"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i><a class="txt" href="waybel_6.html#E8:53"><i><font color="Green">E37</font></i></a>, <a class="ref" href="funct_1.html#T118">FUNCT_1:118</a></i>;<br/>
<a NAME="E4:53_5"/><i><font color="Green">E42</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 1 </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 1 </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1 </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</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="E1:53_5_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>9</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="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 1 </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 1 </span>}</span> 
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_5_1_1"/>
( <font color="Maroon">c<sub>9</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="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 1 </span>}</span>  or <font color="Maroon">c<sub>9</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="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 1 </span>}</span>  )
 </span><b>by </b><i><a class="txt" href="waybel_6.html#E1:53_5_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_5_1_1_1"><b>suppose </b></a><a NAME="E1:53_5_1_1_1"/>
<font color="Maroon">c<sub>9</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="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 1 </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E3:53_5_1_1_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 1 )
 ;<br/><a NAME="E4:53_5_1_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>11</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="E6:53_5_1_1_1"/><i><font color="Green">E45</font></i>: 
(  <a href="funct_3.html#K5">chi</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> is <a href="waybel_6.html#V5" target="_self">prime</a> )
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <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#K3">BoolePoset</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <b>by </b><i><a class="txt" href="waybel_6.html#E6:53_5_1_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="waybel_6.html#T31" target="_self">Th31</a></i>;<br/><a NAME="E8:53_5_1_1_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E9:53_5_1_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="ref" href="funct_1.html#T118">FUNCT_1:118</a></i>;<br/><a NAME="E10:53_5_1_1_1"/><i><font color="Green">E47</font></i>: 
1 <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K4">Top</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="yellow_1.html#T19">YELLOW_1:19</a></i>;<br/><a NAME="E11:53_5_1_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K4">Top</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T45">YELLOW_0:45</a></i>;<br/><a NAME="E12:53_5_1_1_1"/>
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/>( <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  iff <font color="Olive">b<sub>1</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="waybel_6.html#E6:53_5_1_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="waybel_6.html#T32" target="_self">Th32</a></i>;<br/><a NAME="E13:53_5_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> is <a href="waybel_0.html#V20">join-preserving</a>
 <b>by </b><i><a class="txt" href="waybel_6.html#E6:53_5_1_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="waybel_6.html#T25" target="_self">Th25</a></i>;<br/><a NAME="E14:53_5_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>12</sub></font> <a href="waybel_0.html#R4">preserves_sup_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K7">}</a></span> &amp;  <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K7">}</a></span>,<font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K7">}</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="domain_1.html#K7">}</a></span> )
 <b>by </b><i><a class="ref" href="waybel_0.html#D35">WAYBEL_0:def 35</a>, <a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a>, <a class="ref" href="yellow_0.html#T41">YELLOW_0:41</a></i>;<br/><b>then </b><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="waybel_6.html#E9:53_5_1_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="waybel_6.html#E3:53_5_1_1_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="waybel_6.html#E10:53_5_1_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="waybel_6.html#E11:53_5_1_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="yellow_0.html#T24">YELLOW_0:24</a></i>
;<br/><b>hence </b><a NAME="E16:53_5_1_1_1"/>
<font color="Maroon">c<sub>9</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="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1 </span>}</span> 
 <b>by </b><i><a class="txt" href="waybel_6.html#E3:53_5_1_1_1"><i><font color="Green">E44</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_5_1_1_2"><b>suppose </b></a><a NAME="E1:53_5_1_1_2"/>
<font color="Maroon">c<sub>9</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="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 1 </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E3:53_5_1_1_2"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 1 )
 ;<br/><a NAME="E4:53_5_1_1_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>11</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="E6:53_5_1_1_2"/><i><font color="Green">E45</font></i>: 
(  <a href="funct_3.html#K5">chi</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> is <a href="waybel_6.html#V5" target="_self">prime</a> )
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <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#K3">BoolePoset</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <b>by </b><i><a class="txt" href="waybel_6.html#E6:53_5_1_1_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="waybel_6.html#T31" target="_self">Th31</a></i>;<br/><a NAME="E8:53_5_1_1_2"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E9:53_5_1_1_2"/><b>then </b><i><font color="Green">E46</font></i>: 
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="ref" href="funct_1.html#T118">FUNCT_1:118</a></i>;<br/><a NAME="E10:53_5_1_1_2"/><i><font color="Green">E47</font></i>: 
1 <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K4">Top</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="yellow_1.html#T19">YELLOW_1:19</a></i>;<br/><a NAME="E11:53_5_1_1_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K4">Top</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T45">YELLOW_0:45</a></i>;<br/><a NAME="E12:53_5_1_1_2"/>
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/>( <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  iff <font color="Olive">b<sub>1</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="waybel_6.html#E6:53_5_1_1_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="waybel_6.html#T32" target="_self">Th32</a></i>;<br/><a NAME="E13:53_5_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> is <a href="waybel_0.html#V20">join-preserving</a>
 <b>by </b><i><a class="txt" href="waybel_6.html#E6:53_5_1_1_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="waybel_6.html#T25" target="_self">Th25</a></i>;<br/><a NAME="E14:53_5_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>12</sub></font> <a href="waybel_0.html#R4">preserves_sup_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K7">}</a></span> &amp;  <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K7">}</a></span>,<font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K7">}</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="domain_1.html#K7">}</a></span> )
 <b>by </b><i><a class="ref" href="waybel_0.html#D35">WAYBEL_0:def 35</a>, <a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a>, <a class="ref" href="yellow_0.html#T41">YELLOW_0:41</a></i>;<br/><b>then </b><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="waybel_6.html#E9:53_5_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="waybel_6.html#E3:53_5_1_1_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="waybel_6.html#E10:53_5_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="waybel_6.html#E11:53_5_1_1_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="yellow_0.html#T24">YELLOW_0:24</a></i>
;<br/><b>hence </b><a NAME="E16:53_5_1_1_2"/>
<font color="Maroon">c<sub>9</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="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1 </span>}</span> 
 <b>by </b><i><a class="txt" href="waybel_6.html#E3:53_5_1_1_2"><i><font color="Green">E44</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E5:53_5"/><i><font color="Green">E43</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1 </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 1 </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 1 </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_5_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</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="E1:53_5_2"/>
<font color="Maroon">c<sub>9</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="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1 </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E3:53_5_2"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1 )
 ;<br/>
<a NAME="E4:53_5_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>11</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="E6:53_5_2"/><i><font color="Green">E45</font></i>: 
(  <a href="funct_3.html#K5">chi</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> is <a href="waybel_6.html#V5" target="_self">prime</a> )
 ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <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#K3">BoolePoset</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <b>by </b><i><a class="txt" href="waybel_6.html#E6:53_5_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="waybel_6.html#T31" target="_self">Th31</a></i>;<br/>
<a NAME="E8:53_5_2"/><i><font color="Green">E46</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<i><font color="Green">E47</font></i>: the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> = 
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="zfmisc_1.html#K1">bool</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="yellow_1.html#T4">YELLOW_1:4</a></i>
<br/>.= 
 <a href="zfmisc_1.html#K1">bool</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>
<b>by </b><i><a class="ref" href="zfmisc_1.html#T30">ZFMISC_1:30</a></i>
;<br/>
<a NAME="E10:53_5_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i><a class="txt" href="waybel_6.html#E8:53_5_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="funct_1.html#T118">FUNCT_1:118</a></i>;<br/>
<a NAME="E11:53_5_2"/><i><font color="Green">E49</font></i>: 
( ( <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> ) &amp; ( <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> ) )
 
<b>by </b><i><a class="txt" href="waybel_6.html#E9:53_5_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E12:53_5_2"/><i><font color="Green">E50</font></i>: 
1 <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K4">Top</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="yellow_1.html#T19">YELLOW_1:19</a></i>;<br/>
<a NAME="E13:53_5_2"/>
<font color="Maroon">c<sub>12</sub></font> is <a href="waybel_0.html#V20">join-preserving</a>
 
<b>by </b><i><a class="txt" href="waybel_6.html#E6:53_5_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="waybel_6.html#T33" target="_self">Th33</a></i>;<br/>
<a NAME="E14:53_5_2"/><b>then </b><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="waybel_0.html#R4">preserves_sup_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K7">}</a></span> &amp;  <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K7">}</a></span>,<font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K7">}</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="domain_1.html#K7">}</a></span> )
 
<b>by </b><i><a class="ref" href="waybel_0.html#D35">WAYBEL_0:def 35</a>, <a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a>, <a class="ref" href="yellow_0.html#T41">YELLOW_0:41</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_5_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:53_5_2_2"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><b>then </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> = 
<a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#K2">\/</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="yellow_1.html#T17">YELLOW_1:17</a></i>
<br/>.= 
0

;<br/><b>hence </b><a NAME="E3:53_5_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel_6.html#E3:53_5_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="waybel_6.html#E10:53_5_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="waybel_6.html#E14:53_5_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a></i>;<br/></div><b>end;</b></div>
<a NAME="E16:53_5_2"/><b>then </b>
( <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K4">Top</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> or <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K4">Top</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> )
 
<b>by </b><i><a class="txt" href="waybel_6.html#E11:53_5_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="yellow_1.html#T19">YELLOW_1:19</a></i>;<br/>
<a NAME="E17:53_5_2"/><b>then </b>
( <font color="Maroon">c<sub>9</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="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 1 </span>}</span>  or <font color="Maroon">c<sub>9</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="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 1 </span>}</span>  )
 
<b>by </b><i><a class="txt" href="waybel_6.html#E3:53_5_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="waybel_6.html#E12:53_5_2"><i><font color="Green">E50</font></i></a></i>;<br/>
<b>hence </b><a NAME="E18:53_5_2"/>
<font color="Maroon">c<sub>9</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="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 1 </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 1 </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b> <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K7">}</a></span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="waybel_6.html#E3:53_5"><i><font color="Green">E41</font></i></a>, <a class="ref" href="yellow_0.html#T41">YELLOW_0:41</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_1.html#T17">YELLOW_1:17</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 1 </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="waybel_6.html#E9:53"><i><font color="Green">E38</font></i></a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 1 </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 1 </span>}</span> 
<b>by </b><i><a class="txt" href="waybel_6.html#E9:53"><i><font color="Green">E38</font></i></a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1 </span>}</span> 
<b>by </b><i><a class="txt" href="waybel_6.html#E4:53_5"><i><font color="Green">E42</font></i></a>, <a class="txt" href="waybel_6.html#E5:53_5"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="waybel_6.html#E9:53"><i><font color="Green">E38</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K7">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="yellow_0.html#T41">YELLOW_0:41</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E14:53"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/>

<b>assume </b><a NAME="E1:53_6"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<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 <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</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>1</sub></font> <b>by </b><i><a class="txt" href="waybel_6.html#E1:53_6"><i><font color="Green">E42</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_6_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:53_6_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><a NAME="E2:53_6_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 1 </span>}</span> 
 <b>by </b><i><a class="txt" href="waybel_6.html#E9:53"><i><font color="Green">E38</font></i></a></i>;<br/><a NAME="E3:53_6_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> 1 </span>}</span> 
 <b>by </b><i><a class="txt" href="waybel_6.html#E9:53"><i><font color="Green">E38</font></i></a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_6_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_6_1_1"/>
(  not <font color="Maroon">c<sub>9</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> or  not <font color="Maroon">c<sub>10</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> )
 </span><b>by </b><i><a class="txt" href="waybel_6.html#E1:53_6_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="orders_2.html#T25">ORDERS_2:25</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_6_1_1_1"><b>suppose </b></a><a NAME="E1:53_6_1_1_1"/>
not <font color="Maroon">c<sub>9</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>11</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="E3:53_6_1_1_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_6.html#K4" target="_self">PRIME</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>11</sub></font> &amp; not <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="waybel_6.html#E1:53"><i><font color="Green">E34</font></i></a>, <a class="ref" href="waybel_6.html#T17" target="_self">Th17</a></i>;<br/><b>set </b><font color="Maroon">c<sub>12</sub></font> =  <a href="funct_3.html#K5">chi</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E4:53_6_1_1_1"/>
<font color="Maroon">c<sub>11</sub></font> is <a href="waybel_6.html#V5" target="_self">prime</a>
 <b>by </b><i><a class="txt" href="waybel_6.html#E3:53_6_1_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="waybel_6.html#D7" target="_self">Def7</a></i>;<br/><a NAME="E5:53_6_1_1_1"/><b>then </b>
 <a href="funct_3.html#K5">chi</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> =  <a href="funct_3.html#K5">chi</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> ;<br/><a NAME="E7:53_6_1_1_1"/><i><font color="Green">E47</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> )
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/><a NAME="E8:53_6_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E9:53_6_1_1_1"/><b>then </b><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="txt" href="waybel_6.html#E7:53_6_1_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_6_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:53_6_1_1_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><a NAME="E2:53_6_1_1_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="txt" href="waybel_6.html#E3:53_6_1"><i><font color="Green">E45</font></i></a></i>;<br/><b>hence </b><a NAME="E3:53_6_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel_6.html#E3:53_6_1_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="waybel_6.html#T32" target="_self">Th32</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E11:53_6_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel_6.html#E1:53_6"><i><font color="Green">E42</font></i></a>, <a class="txt" href="waybel_6.html#E2:53_6_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="waybel_6.html#E3:53_6_1_1_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="waybel_6.html#E9:53_6_1_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="waybel_6.html#T32" target="_self">Th32</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_6_1_1_2"><b>suppose </b></a><a NAME="E1:53_6_1_1_2"/>
not <font color="Maroon">c<sub>10</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>11</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="E3:53_6_1_1_2"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_6.html#K4" target="_self">PRIME</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>11</sub></font> &amp; not <font color="Maroon">c<sub>10</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="waybel_6.html#E1:53"><i><font color="Green">E34</font></i></a>, <a class="ref" href="waybel_6.html#T17" target="_self">Th17</a></i>;<br/><b>set </b><font color="Maroon">c<sub>12</sub></font> =  <a href="funct_3.html#K5">chi</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E4:53_6_1_1_2"/>
<font color="Maroon">c<sub>11</sub></font> is <a href="waybel_6.html#V5" target="_self">prime</a>
 <b>by </b><i><a class="txt" href="waybel_6.html#E3:53_6_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="waybel_6.html#D7" target="_self">Def7</a></i>;<br/><a NAME="E5:53_6_1_1_2"/><b>then </b>
 <a href="funct_3.html#K5">chi</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> =  <a href="funct_3.html#K5">chi</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> ;<br/><a NAME="E7:53_6_1_1_2"/><i><font color="Green">E47</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> )
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/><a NAME="E8:53_6_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E9:53_6_1_1_2"/><b>then </b><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="txt" href="waybel_6.html#E7:53_6_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_6_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:53_6_1_1_2_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><a NAME="E2:53_6_1_1_2_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="txt" href="waybel_6.html#E2:53_6_1"><i><font color="Green">E44</font></i></a></i>;<br/><b>hence </b><a NAME="E3:53_6_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel_6.html#E3:53_6_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="waybel_6.html#T32" target="_self">Th32</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E11:53_6_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel_6.html#E1:53_6"><i><font color="Green">E42</font></i></a>, <a class="txt" href="waybel_6.html#E3:53_6_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="waybel_6.html#E3:53_6_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="waybel_6.html#E9:53_6_1_1_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="waybel_6.html#T32" target="_self">Th32</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:53_6"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E15:53"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="waybel_1.html#V2">distributive</a>
 <b>by </b><i><a class="txt" href="waybel_6.html#E12:53"><i><font color="Green">E39</font></i></a>, <a class="txt" href="waybel_6.html#E13:53"><i><font color="Green">E40</font></i></a>, <a class="ref" href="waybel_6.html#T3" target="_self">Th3</a></i>;<br/>

<a NAME="E16:53"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="waybel_0.html#V18">sups-preserving</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D33">WAYBEL_0:def 33</a><br/>

<a NAME="E1:53_7"/>
<font color="Maroon">c<sub>6</sub></font> <a href="waybel_0.html#R4">preserves_sup_of</a> <font color="Maroon">c<sub>7</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_7_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:53_7_1"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a><br/>

<b>thus </b><a NAME="E2:53_7_1"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>6</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>7</sub></font>, <a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>

<a NAME="E3:53_7_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1 </span>}</span> 
 
<b>by </b><i><a class="txt" href="waybel_6.html#E9:53"><i><font color="Green">E38</font></i></a></i>;<br/>
<a NAME="E4:53_7_1"/><i><font color="Green">E43</font></i>: 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_7_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</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="E1:53_7_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:53_7_1_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T21">YELLOW_1:21</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:53_7_1_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:53_7_1_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_6.html#E4:53_7_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</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 <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="waybel_6.html#E6:53_7_1_1"><i><font color="Green">E45</font></i></a></i>;<br/>
<a NAME="E8:53_7_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> 1 </span>}</span> 
 
<b>by </b><i><a class="txt" href="waybel_6.html#E9:53"><i><font color="Green">E38</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E10:53_7_1_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="txt" href="waybel_6.html#E4:53_7_1_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="waybel_6.html#E6:53_7_1_1"><i><font color="Green">E45</font></i></a></i>;<br/>
<a NAME="E11:53_7_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>13</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="E13:53_7_1_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="funct_3.html#K5">chi</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> is <a href="waybel_6.html#V5" target="_self">prime</a> )
 ;<br/>
<a NAME="E14:53_7_1_1"/><i><font color="Green">E48</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> )
 
<b>by </b><i><a class="txt" href="waybel_6.html#E13:53_7_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E15:53_7_1_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E16:53_7_1_1"/><b>then </b><i><font color="Green">E49</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="txt" href="waybel_6.html#E14:53_7_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_7_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:53_7_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:53_7_1_1_1"/><b>then </b><i><font color="Green">E50</font></i>: 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>7</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_6.html#E13:53_7_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="waybel_6.html#T32" target="_self">Th32</a></i>;<br/><a NAME="E3:53_7_1_1_1"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>7</sub></font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T32">YELLOW_0:32</a></i>;<br/><a NAME="E4:53_7_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>11</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>7</sub></font>
 <b>by </b><i><a class="txt" href="waybel_6.html#E6:53_7_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/><a NAME="E5:53_7_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>11</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_6.html#E2:53_7_1_1_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/><b>hence </b><a NAME="E6:53_7_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel_6.html#E10:53_7_1_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="waybel_6.html#E13:53_7_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="waybel_6.html#T32" target="_self">Th32</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E18:53_7_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_6.html#E3:53_7_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="waybel_6.html#E10:53_7_1_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="waybel_6.html#E16:53_7_1_1"><i><font color="Green">E49</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:53_7_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_7_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</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="E1:53_7_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E3:53_7_1_2"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>8</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> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="txt" href="waybel_6.html#E3:53_7_1"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E4:53_7_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 
;<br/>
<b>then 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="E6:53_7_1_2"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="funct_3.html#K5">chi</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> is <a href="waybel_6.html#V5" target="_self">prime</a> )
 ;<br/>
<a NAME="E7:53_7_1_2"/><i><font color="Green">E46</font></i>: 
not  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>7</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_6.html#E3:53_7_1_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="waybel_6.html#E6:53_7_1_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="waybel_6.html#T32" target="_self">Th32</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_7_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:53_7_1_2_1"/>
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> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><a NAME="E2:53_7_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/><b>hence </b><a NAME="E3:53_7_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel_6.html#E7:53_7_1_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="yellow_0.html#T32">YELLOW_0:32</a></i>;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>11</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="E10:53_7_1_2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; not <font color="Maroon">c<sub>11</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> )
 ;<br/>
<a NAME="E11:53_7_1_2"/><i><font color="Green">E48</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> )
 
<b>by </b><i><a class="txt" href="waybel_6.html#E6:53_7_1_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E12:53_7_1_2"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E13:53_7_1_2"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="txt" href="waybel_6.html#E11:53_7_1_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E14:53_7_1_2"/><b>then </b>
<font color="Maroon">c<sub>9</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="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> 1 </span>}</span> 
 
<b>by </b><i><a class="txt" href="waybel_6.html#E6:53_7_1_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="waybel_6.html#E10:53_7_1_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="waybel_6.html#T32" target="_self">Th32</a></i>;<br/>
<a NAME="E15:53_7_1_2"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_6.html#E9:53"><i><font color="Green">E38</font></i></a></i>;<br/>
<a NAME="E16:53_7_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_6.html#E8:53"><i><font color="Green">E37</font></i></a>, <a class="txt" href="waybel_6.html#E10:53_7_1_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E17:53_7_1_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_6.html#E3:53_7_1_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="waybel_6.html#E15:53_7_1_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E18:53_7_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_1.html#T21">YELLOW_1:21</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:53_7_1"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_6.html#E4:53_7_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:53_7"/>
<font color="Maroon">c<sub>6</sub></font> <a href="waybel_0.html#R4">preserves_sup_of</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E17:53"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="waybel_2.html#V2">meet-continuous</a>
 <b>by </b><i><a class="txt" href="waybel_6.html#E12:53"><i><font color="Green">E39</font></i></a>, <a class="txt" href="waybel_6.html#E14:53"><i><font color="Green">E41</font></i></a>, <a class="ref" href="waybel_6.html#T4" target="_self">Th4</a></i>;<br/>


</div>
