<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">L</font> be  <a href="yellow_0.html#V1">lower-bounded</a> <a href="lattice3.html#V1">with_suprema</a> <a href="orders_2.html#NM1">Poset</a>;<br/><b>let </b><font color="Maroon">a</font> be  <a href="waybel_4.html#V3" target="_self">auxiliary(iii)</a> <a href="alg_1.html#NM1">Relation</a> of <font color="Maroon">L</font>, <font color="Maroon">b</font> be  <a href="waybel_4.html#V3" target="_self">auxiliary(iii)</a> <a href="alg_1.html#NM1">Relation</a> of <font color="Maroon">L</font>;<br/>



<b>reconsider </b><font color="Maroon">ab</font> = <font color="Maroon">a</font> <a href="relset_1.html#K2">/\</a> <font color="Maroon">b</font> as   <a href="alg_1.html#NM1">Relation</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E2:28"/>
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">z</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">ab</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">z</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">ab</font> holds <br/><span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">x</font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">y</font></span>)</span>,<font color="Olive">z</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">ab</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>proof </b></a><div class="add">

<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>, <font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<b>assume </b><a NAME="E1:28_1"/>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">ab</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">ab</font> )
 ;<br/>

<a NAME="E2:28_1"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:28_1"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:28_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">ab</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="E3:28"/>
<font color="Maroon">a</font> <a href="relset_1.html#K2">/\</a> <font color="Maroon">b</font> is  <a href="waybel_4.html#V3" target="_self">auxiliary(iii)</a> <a href="alg_1.html#NM1">Relation</a> of <font color="Maroon">L</font>
 <b>by </b><i/>;<br/>


</div>
