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<b>let </b><font color="Maroon">L</font> be  <a href="orders_2.html#V3">transitive</a> <a href="orders_2.html#V4">antisymmetric</a> <a href="lattice3.html#V1">with_suprema</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">V</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>





<b>assume </b><a NAME="E1:31"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font>
 ;<br/>

<b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_4.html#D1">YELLOW_4:def 1</a><br/>

<b>assume </b><a NAME="E2:31"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K2">"\/"</a> <font color="Maroon">V</font>
 ;<br/>

<a NAME="E3:31"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K2">"\/"</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">s</font></span>)</span> where <font color="Olive">s</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="yellow_4.html#T15">YELLOW_4:15</a></i>;<br/>
<b>then consider </b><font color="Maroon">s</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E5:31"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E6:31"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i/>;<br/>
<b>take </b><font color="Maroon">a</font> = <font color="Maroon">y</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">s</font>;<br/>

<a NAME="E7:31"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K2">"\/"</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">t</font></span>)</span> where <font color="Olive">t</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="yellow_4.html#T15">YELLOW_4:15</a></i>;<br/>
<b>hence </b><a NAME="E8:31"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K2">"\/"</a> <font color="Maroon">V</font>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E9:31"/>
<font color="Maroon">b</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="txt" href="waybel12.html#E1"><i><font color="Green">Lemma18</font></i></a>, <a class="ref" href="waybel_1.html#T3">WAYBEL_1:3</a></i>;<br/>


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