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<div class="add">

<b>let </b><font color="Maroon">L</font> be  <a href="yellow_0.html#V1">lower-bounded</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">R</font> be  <a href="waybel_4.html#V5">auxiliary</a> <a href="alg_1.html#NM1">Relation</a> of <font color="Maroon">L</font>;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:59_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">R</font> is <a href="waybel_7.html#V5" target="_self">multiplicative</a>
 ;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>thus </b><a NAME="E2:59_1"/>
<font color="Maroon">R</font> <a href="waybel_4.html#K6">-above</a> <font color="Maroon">x</font> is <a href="waybel_0.html#V2">filtered</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">z</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="waybel_0.html#D2">WAYBEL_0:def 2</a><br/>

<b>assume </b><a NAME="E1:59_1_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="waybel_4.html#K6">-above</a> <font color="Maroon">x</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="waybel_4.html#K6">-above</a> <font color="Maroon">x</font> )
 ;<br/>

<a NAME="E2:59_1_1"/><b>then </b>
( <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 
<b>by </b><i><a class="ref" href="waybel_4.html#T14">WAYBEL_4:14</a></i>;<br/>
<a NAME="E3:59_1_1"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z</font></span>)</span></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:59_1_1"/><b>then </b>
( <font color="Maroon">y</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="waybel_4.html#K6">-above</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z</font> &amp; <font color="Maroon">z</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z</font> )
 
<b>by </b><i><a class="ref" href="waybel_4.html#T14">WAYBEL_4:14</a>, <a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/>
<b>hence </b><a NAME="E5:59_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="waybel_4.html#K6">-above</a> <font color="Maroon">x</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">z</font> )
 ;<br/>


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

<b>assume </b><a NAME="E2:59"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <font color="Maroon">R</font> <a href="waybel_4.html#K6">-above</a> <font color="Olive">x</font> is <a href="waybel_0.html#V2">filtered</a>
 ;<br/>

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <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>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_7.html#D7" target="_self">WAYBEL_7:def 7</a><br/>

<b>assume </b><a NAME="E3:59"/>
( <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">x</font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">y</font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 ;<br/>

<a NAME="E4:59"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="waybel_4.html#K6">-above</a> <font color="Maroon">a</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="waybel_4.html#K6">-above</a> <font color="Maroon">a</font> &amp; <font color="Maroon">R</font> <a href="waybel_4.html#K6">-above</a> <font color="Maroon">a</font> is <a href="waybel_0.html#V2">filtered</a> )
 
<b>by </b><i>, <a class="ref" href="waybel_4.html#T14">WAYBEL_4:14</a></i>;<br/>
<a NAME="E5:59"/><b>then </b>
<font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="waybel_4.html#K6">-above</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T41">WAYBEL_0:41</a></i>;<br/>
<b>hence </b><a NAME="E6:59"/>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font></span>)</span></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="waybel_4.html#T14">WAYBEL_4:14</a></i>;<br/>


</div>
