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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="waybel_0.html#NM6">Semilattice</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:7"/><i><font color="Green">E4</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="waybel_0.html#V19">meet-preserving</a>
 <b>and </b><br/><a NAME="E2:7"/><i><font color="Green">E5</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V2">filtered</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>3</sub></font> <a href="waybel_0.html#R3">preserves_inf_of</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E3:7"/><i><font color="Green">E6</font></i>: 
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">c<sub>4</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#D30">WAYBEL_0:def 30</a><br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>4</sub></font> st <br/>(  <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> );<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:7"/><i><font color="Green">E7</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> iff ( <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> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E8:7"/><i><font color="Green">E8</font></i>: 
(  <a href="yellow_0.html#R2">ex_inf_of</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">c<sub>1</sub></font> &amp;  <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T38">YELLOW_0:38</a>, <a class="ref" href="yellow_0.html#T39">YELLOW_0:39</a></i>;<br/>
<a NAME="E9:7"/>
<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 <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_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>thus </b><a NAME="E1:7_1"/>
(  not <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>8</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> )
 <b>by </b><i><a class="txt" href="waybel21.html#E5:7"><i><font color="Green">E7</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>5</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="waybel21.html#E5:7"><i><font color="Green">E7</font></i></a>, <a class="txt" href="waybel21.html#E8:7"><i><font color="Green">E8</font></i></a></i>;<br/>
<div><i><font color="Green">E9</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>4</sub></font>;<br/><a NAME="E1:7_2"/>
<font color="Maroon">c<sub>9</sub></font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E2:7_2"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  implies  <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_0.html#T55">YELLOW_0:55</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E10</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="7_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>4</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>assume </b><a NAME="E2:7_3"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E3:7_3"/><b>then </b>
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T55">YELLOW_0:55</a></i>;<br/><b>hence </b><a NAME="E4:7_3"/>
 <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="waybel21.html#E5:7"><i><font color="Green">E7</font></i></a>, <a class="txt" href="waybel21.html#E2:7_3"><i><font color="Green">E11</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:7_4"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><a NAME="E2:7_4"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>4</sub></font> st <br/>(  <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="waybel21.html#E5:7"><i><font color="Green">E7</font></i></a></i>;<br/><b>hence </b><a NAME="E3:7_4"/>
ex <font color="Olive">b<sub>1</sub></font> being <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>4</sub></font> st <br/>(  <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> )
 ;<br/></div><b>end;</b></div>
<a NAME="E14:7"/><b>then </b><i><font color="Green">E11</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> is <a href="waybel_0.html#V2">filtered</a> &amp;  <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp;  <a href="yellow_0.html#NK4" target="_self">inf</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> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="txt" href="waybel21.html#E3:7"><i><font color="Green">E6</font></i></a>, <a class="txt" href="waybel21.html#E11:7"><i><font color="Green">E9</font></i></a>, <a class="txt" href="waybel21.html#E12:7"><i><font color="Green">E10</font></i></a>, <a class="ref" href="waybel_0.html#T56">WAYBEL_0:56</a>, <a class="ref" href="waybel_0.html#T58">WAYBEL_0:58</a>, <a class="ref" href="waybel_0.html#T59">WAYBEL_0:59</a></i>;<br/>
<a NAME="E15:7"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="waybel_0.html#R3">preserves_inf_of</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="waybel21.html#E2:7"><i><font color="Green">E5</font></i></a></i>;<br/>
<a NAME="E16:7"/><b>then </b><i><font color="Green">E12</font></i>: 
(  <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp;  <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="yellow_6.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> &amp;  <a href="yellow_0.html#NK4" target="_self">inf</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> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="waybel21.html#E14:7"><i><font color="Green">E11</font></i></a>, <a class="ref" href="waybel_0.html#D30">WAYBEL_0:def 30</a></i>;<br/>
<a NAME="E17:7"/>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_5"><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="E1:7_5"/><i><font color="Green">E13</font></i>: 
<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 reconsider </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K1">}</a></span> as  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</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="waybel21.html#E1:7_5"><i><font color="Green">E13</font></i></a></i>;<br/>
<a NAME="E4:7_5"/>
<font color="Maroon">c<sub>10</sub></font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E5:7_5"/><b>then </b>
(  <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T39">YELLOW_0:39</a>, <a class="ref" href="yellow_0.html#T55">YELLOW_0:55</a></i>;<br/>
<b>hence </b><a NAME="E6:7_5"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="waybel21.html#E5:7"><i><font color="Green">E7</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:7"/><b>then </b><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T156">RELAT_1:156</a></i>;<br/>
<a NAME="E19:7"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>8</sub></font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Maroon">c<sub>3</sub></font> <a href="yellow_6.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel21.html#E16:7"><i><font color="Green">E12</font></i></a>, <a class="ref" href="yellow_0.html#T31">YELLOW_0:31</a></i>;<br/>
<a NAME="E20:7"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Maroon">c<sub>3</sub></font> <a href="yellow_6.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a><br/>

<b>assume </b><a NAME="E1:7_6"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>hence </b><a NAME="E2:7_6"/>
<font color="Maroon">c<sub>3</sub></font> <a href="yellow_6.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="waybel21.html#E18:7"><i><font color="Green">E13</font></i></a>, <a class="txt" href="waybel21.html#E19:7"><i><font color="Green">E14</font></i></a>, <a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E16</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="7_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>assume </b><a NAME="E1:7_7"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><a NAME="E2:7_7"/>
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>8</sub></font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Maroon">c<sub>9</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_7_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a><br/>

<b>assume </b><a NAME="E1:7_7_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:7_7_1"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>12</sub></font> being  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E5:7_7_1"/><i><font color="Green">E19</font></i>: 
(  <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="waybel21.html#E5:7"><i><font color="Green">E7</font></i></a>, <a class="txt" href="waybel21.html#E3:7_7_1"><i><font color="Green">E18</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E7:7_7_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="waybel_0.html#R3">preserves_inf_of</a> <font color="Maroon">c<sub>13</sub></font> )
 
<b>by </b><i><a class="txt" href="waybel21.html#E1:7"><i><font color="Green">E4</font></i></a>, <a class="ref" href="waybel21.html#T2" target="_self">Th2</a>, <a class="ref" href="relat_1.html#T156">RELAT_1:156</a></i>;<br/>
<a NAME="E8:7_7_1"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> &amp;  <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="waybel21.html#E1:7_7"><i><font color="Green">E17</font></i></a>, <a class="txt" href="waybel21.html#E3:7_7_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="waybel21.html#E5:7_7_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="waybel_0.html#D30">WAYBEL_0:def 30</a>, <a class="ref" href="yellow_0.html#T9">YELLOW_0:9</a></i>;<br/>
<b>hence </b><a NAME="E9:7_7_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="yellow_0.html#D10">YELLOW_0:def 10</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:7_7"/>
<font color="Maroon">c<sub>3</sub></font> <a href="yellow_6.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="waybel21.html#E16:7"><i><font color="Green">E12</font></i></a>, <a class="ref" href="yellow_0.html#T31">YELLOW_0:31</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E22:7"/>
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="waybel21.html#E20:7"><i><font color="Green">E15</font></i></a>, <a class="ref" href="yellow_0.html#T16">YELLOW_0:16</a></i>;<br/>

<b>hence </b><a NAME="E23:7"/>
 <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="yellow_6.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel21.html#E20:7"><i><font color="Green">E15</font></i></a>, <a class="txt" href="waybel21.html#E21:7"><i><font color="Green">E16</font></i></a>, <a class="ref" href="yellow_0.html#D10">YELLOW_0:def 10</a></i>;<br/>


</div>
