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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="lattice3.html#V2">with_infima</a> <a href="orders_2.html#NM1">Poset</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:6"/><i><font color="Green">E4</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E2:6"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>thus </b><a NAME="E3:6"/>
<font color="Maroon">c<sub>2</sub></font> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> <a href="yellow_4.html#K2">"\/"</a> <font color="Maroon">c<sub>5</sub></font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:6_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:6_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, 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> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="yellow_4.html#D4">YELLOW_4:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:6_1"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E5:6_1"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E6:6_1"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>
<a NAME="E7:6_1"/><i><font color="Green">E9</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="lattice3.html#K8">~</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="lattice3.html#K8">~</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="waybel16.html#E1:6"><i><font color="Green">E4</font></i></a>, <a class="txt" href="waybel16.html#E2:6"><i><font color="Green">E5</font></i></a>, <a class="txt" href="waybel16.html#E5:6_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="waybel16.html#E6:6_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="lattice3.html#D6">LATTICE3:def 6</a></i>;<br/>
<a NAME="E8:6_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattice3.html#K8">~</a> </span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="lattice3.html#K8">~</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="waybel16.html#E4:6_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="yellow_7.html#T21">YELLOW_7:21</a></i>;<br/>
<a NAME="E9:6_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span>, B is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="waybel16.html#E7:6_1"><i><font color="Green">E9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:6_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="yellow_4.html#K2">"\/"</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="yellow_4.html#D3">YELLOW_4:def 3</a></i>;<br/>


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

<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="E4:6"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="yellow_4.html#K2">"\/"</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E5:6"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span>, B is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="yellow_4.html#D3">YELLOW_4:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span><b> such that </b><br/><a NAME="E7:6"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E8:6"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E9:6"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>
<a NAME="E10:6"/><i><font color="Green">E9</font></i>: 
(  <a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="waybel16.html#E1:6"><i><font color="Green">E4</font></i></a>, <a class="txt" href="waybel16.html#E2:6"><i><font color="Green">E5</font></i></a>, <a class="txt" href="waybel16.html#E8:6"><i><font color="Green">E7</font></i></a>, <a class="txt" href="waybel16.html#E9:6"><i><font color="Green">E8</font></i></a>, <a class="ref" href="lattice3.html#D7">LATTICE3:def 7</a></i>;<br/>
<a NAME="E11:6"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel16.html#E7:6"><i><font color="Green">E6</font></i></a>, <a class="ref" href="yellow_7.html#T22">YELLOW_7:22</a></i>;<br/>
<a NAME="E12:6"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, 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> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="waybel16.html#E10:6"><i><font color="Green">E9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E13:6"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="yellow_4.html#D4">YELLOW_4:def 4</a></i>;<br/>


</div>
