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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V4">antisymmetric</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/>

<b>assume </b><a NAME="E1:84"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="lattice3.html#V1">with_suprema</a> or <font color="Maroon">c<sub>1</sub></font> is <a href="lattice3.html#V2">with_infima</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>assume </b><b>that </b><br/><a NAME="E2:84"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is <a href="waybel_0.html#V12" target="_self">lower</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="waybel_0.html#V1" target="_self">directed</a> )
 <b>and </b><br/><a NAME="E3:84"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> is <a href="waybel_0.html#V12" target="_self">lower</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="waybel_0.html#V1" target="_self">directed</a> )
 ;<br/>

<a NAME="E4:84"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font> is <a href="waybel_0.html#V12" target="_self">lower</a>
 
<b>by </b><i><a class="txt" href="waybel_0.html#E2:84"><i><font color="Green">E44</font></i></a>, <a class="txt" href="waybel_0.html#E3:84"><i><font color="Green">E45</font></i></a>, <a class="ref" href="waybel_0.html#T27" target="_self">Th27</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:84_1"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="lattice3.html#V1">with_suprema</a> or <font color="Maroon">c<sub>1</sub></font> is <a href="lattice3.html#V2">with_infima</a> )
 </span><b>by </b><i><a class="txt" href="waybel_0.html#E1:84"><i><font color="Green">E43</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_1_1"><b>suppose </b></a><a NAME="E1:84_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="lattice3.html#V1">with_suprema</a>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_1_1"><b>now </b></a><div class="add"><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#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:84_1_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><a NAME="E2:84_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:84_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#K10">"\/"</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#K10">"\/"</a> <font color="Maroon">c<sub>5</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="waybel_0.html#E2:84"><i><font color="Green">E44</font></i></a>, <a class="txt" href="waybel_0.html#E3:84"><i><font color="Green">E45</font></i></a>, <a class="txt" href="waybel_0.html#E1:84_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="waybel_0.html#T40" target="_self">Th40</a></i>;<br/><b>hence </b><a NAME="E4:84_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#K10">"\/"</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font>
 <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>hence </b><a NAME="E3:84_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font> is <a href="waybel_0.html#V1" target="_self">directed</a>
 <b>by </b><i><a class="txt" href="waybel_0.html#E4:84"><i><font color="Green">E46</font></i></a>, <a class="txt" href="waybel_0.html#E1:84_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="waybel_0.html#T40" target="_self">Th40</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_1_2"><b>suppose </b></a><a NAME="E1:84_1_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="lattice3.html#V2">with_infima</a>
 ;<br/><div class="add"><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#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#D1" target="_self">WAYBEL_0:def 1</a><br/><b>assume </b><a NAME="E2:84_1_2"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><a NAME="E3:84_1_2"/><b>then </b><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 <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>6</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="E5:84_1_2"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_0.html#E2:84"><i><font color="Green">E44</font></i></a>, <a class="ref" href="waybel_0.html#D1" target="_self">Def1</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:84_1_2"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_0.html#E3:84"><i><font color="Green">E45</font></i></a>, <a class="txt" href="waybel_0.html#E3:84_1_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="waybel_0.html#D1" target="_self">Def1</a></i>;<br/><b>take </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="lattice3.html#K11">"/\"</a> <font color="Maroon">c<sub>7</sub></font>;<br/><a NAME="E8:84_1_2"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_0.html#E1:84_1_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/><a NAME="E9:84_1_2"/><b>then </b>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <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="waybel_0.html#E2:84"><i><font color="Green">E44</font></i></a>, <a class="txt" href="waybel_0.html#E3:84"><i><font color="Green">E45</font></i></a>, <a class="txt" href="waybel_0.html#E5:84_1_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="waybel_0.html#E7:84_1_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="waybel_0.html#D19" target="_self">Def19</a></i>;<br/><b>hence </b><a NAME="E10:84_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>thus </b><a NAME="E11:84_1_2"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_0.html#E1:84_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="waybel_0.html#E5:84_1_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="waybel_0.html#E7:84_1_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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