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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="orders_2.html#V4">antisymmetric</a> <a href="lattice3.html#V2" target="_self">with_infima</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</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="E1:41"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="orders_2.html#V3">transitive</a>
 ;<br/>

<a NAME="E2:41"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K11" target="_self">"/\"</a> <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K11" target="_self">"/\"</a> <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K11" target="_self">"/\"</a> <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K11" target="_self">"/\"</a> <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K11" target="_self">"/\"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattice3.html#K11" target="_self">"/\"</a> <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K11" target="_self">"/\"</a> <font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K11" target="_self">"/\"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattice3.html#K11" target="_self">"/\"</a> <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="lattice3.html#E28"><i><font color="Green">Lemma17</font></i></a></i>;<br/>
<a NAME="E3:41"/><b>then </b><i><font color="Green">E24</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K11" target="_self">"/\"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattice3.html#K11" target="_self">"/\"</a> <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K11" target="_self">"/\"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattice3.html#K11" target="_self">"/\"</a> <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="lattice3.html#E1:41"><i><font color="Green">E22</font></i></a>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/>
<a NAME="E4:41"/><b>then </b><i><font color="Green">E25</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K11" target="_self">"/\"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattice3.html#K11" target="_self">"/\"</a> <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K11" target="_self">"/\"</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="lattice3.html#E2:41"><i><font color="Green">E23</font></i></a>, <a class="txt" href="lattice3.html#E28"><i><font color="Green">Lemma17</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>now </b></a><div class="add"><b>let </b><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="E1:41_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K11" target="_self">"/\"</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/><a NAME="E2:41_1"/><b>then </b><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="lattice3.html#E1:41"><i><font color="Green">E22</font></i></a>, <a class="txt" href="lattice3.html#E2:41"><i><font color="Green">E23</font></i></a>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/><a NAME="E3:41_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K11" target="_self">"/\"</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="lattice3.html#E1:41_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="lattice3.html#E28"><i><font color="Green">Lemma17</font></i></a></i>;<br/><b>hence </b><a NAME="E4:41_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K11" target="_self">"/\"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattice3.html#K11" target="_self">"/\"</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="lattice3.html#E2:41_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="lattice3.html#E28"><i><font color="Green">Lemma17</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:41"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K11" target="_self">"/\"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattice3.html#K11" target="_self">"/\"</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K11" target="_self">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K11" target="_self">"/\"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice3.html#E3:41"><i><font color="Green">E24</font></i></a>, <a class="txt" href="lattice3.html#E4:41"><i><font color="Green">E25</font></i></a>, <a class="ref" href="lattice3.html#D14" target="_self">Def14</a></i>;<br/>


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