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<b>let </b><font color="Maroon">L</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>



<b>given </b><font color="Maroon">k</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><a NAME="E1:32"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">p</font> <a href="waybel_4.html#R3">is_maximal_in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">k</font></span>)</span>
 ;<br/>

<a NAME="E2:32"/><i><font color="Green">E26</font></i>: 
<span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">p</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">p</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:32"/><b>then </b><i><font color="Green">E28</font></i>: 
 <a href="yellow_0.html#K2">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">p</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>,<font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">k</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:32"/><i><font color="Green">E29</font></i>: 
 <a href="yellow_0.html#R2">ex_inf_of</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">p</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<a NAME="E5:32"/>
( <font color="Maroon">p</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">p</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">p</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">k</font> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/>
<a NAME="E6:32"/><b>then </b>
( <font color="Maroon">p</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">p</font> &amp; <font color="Maroon">p</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">k</font> )
 
<b>by </b><i><a class="ref" href="waybel_0.html#T18">WAYBEL_0:18</a></i>;<br/>
<a NAME="E7:32"/><b>then </b>
<font color="Maroon">p</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">p</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E8:32"/><b>then </b>
 <a href="waybel_1.html#R1">ex_min_of</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">p</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="waybel16.html#T1" target="_self">Th1</a>, , , <a class="ref" href="waybel_1.html#D4">WAYBEL_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E9:32"/>
<font color="Maroon">p</font> is <a href="waybel16.html#V1" target="_self">completely-irreducible</a>
 <b>by </b><i/>;<br/>


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