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<div class="add">

<b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V2">reflexive</a>  <a href="orders_2.html#L1">RelStr</a> , <font color="Maroon">Y</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V2">reflexive</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/>

<b>assume </b><a NAME="E1:41"/><i><font color="Green">E18</font></i>: 
<span class="p1"><a href="yellow_3.html#K3" target="_self">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="yellow_3.html#K3" target="_self">:]</a></span> is <a href="lattice3.html#V1">with_suprema</a>
 ;<br/>

<a NAME="E2:41"/><i><font color="Green">E20</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3" target="_self">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="yellow_3.html#K3" target="_self">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i/>;<br/>
<b>thus </b><a NAME="E3:41"/>
<font color="Maroon">X</font> is <a href="lattice3.html#V1">with_suprema</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font>;<br/>
<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D10">LATTICE3:def 10</a><br/>

<b>consider </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1"><a href="yellow_3.html#K3" target="_self">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="yellow_3.html#K3" target="_self">:]</a></span><b> such that </b><br/><a NAME="E3:41_1"/><i><font color="Green">E25</font></i>: 
( <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">a</font></span><a href="yellow_3.html#K7" target="_self">]</a></span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> &amp; <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">a</font></span><a href="yellow_3.html#K7" target="_self">]</a></span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> )
 <b>and </b><br/><a NAME="E4:41_1"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">z'</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1"><a href="yellow_3.html#K3" target="_self">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="yellow_3.html#K3" target="_self">:]</a></span>  st <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">a</font></span><a href="yellow_3.html#K7" target="_self">]</a></span> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">z'</font> &amp; <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">a</font></span><a href="yellow_3.html#K7" target="_self">]</a></span> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">z'</font> holds <br/><font color="Maroon">z</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">z'</font>
 <b>by </b><i>, <a class="ref" href="lattice3.html#D10">LATTICE3:def 10</a></i>;<br/>
<b>take </b>
<font color="Maroon">z</font> <a href="yellow_3.html#K8" target="_self">`1</a> 
;<br/>

<a NAME="E5:41_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="yellow_3.html#K8" target="_self">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="yellow_3.html#K9" target="_self">`2</a> </span>)</span></span><a href="yellow_3.html#K7" target="_self">]</a></span>
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/>
<b>hence </b><a NAME="E6:41_1"/>
( <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> <a href="yellow_3.html#K8" target="_self">`1</a>  &amp; <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> <a href="yellow_3.html#K8" target="_self">`1</a>  )
 <b>by </b><i>, </i>;<br/>

<b>let </b><font color="Maroon">z'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E7:41_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z'</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z'</font> )
 ;<br/>

<a NAME="E8:41_1"/>
<font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font>
 
;<br/>
<a NAME="E9:41_1"/><b>then </b>
( <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">a</font></span><a href="yellow_3.html#K7" target="_self">]</a></span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">z'</font>,<font color="Maroon">a</font></span><a href="yellow_3.html#K7" target="_self">]</a></span> &amp; <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">a</font></span><a href="yellow_3.html#K7" target="_self">]</a></span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">z'</font>,<font color="Maroon">a</font></span><a href="yellow_3.html#K7" target="_self">]</a></span> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:41_1"/><b>then </b>
<font color="Maroon">z</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">z'</font>,<font color="Maroon">a</font></span><a href="yellow_3.html#K7" target="_self">]</a></span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E11:41_1"/>
<font color="Maroon">z</font> <a href="yellow_3.html#K8" target="_self">`1</a>  <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z'</font>
 <b>by </b><i>, </i>;<br/>


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

<b>consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/>
<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D10">LATTICE3:def 10</a><br/>

<b>consider </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1"><a href="yellow_3.html#K3" target="_self">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="yellow_3.html#K3" target="_self">:]</a></span><b> such that </b><br/><a NAME="E6:41"/><i><font color="Green">E56</font></i>: 
( <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">x</font></span><a href="yellow_3.html#K7" target="_self">]</a></span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> &amp; <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">y</font></span><a href="yellow_3.html#K7" target="_self">]</a></span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> )
 <b>and </b><br/><a NAME="E7:41"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">z'</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1"><a href="yellow_3.html#K3" target="_self">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="yellow_3.html#K3" target="_self">:]</a></span>  st <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">x</font></span><a href="yellow_3.html#K7" target="_self">]</a></span> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">z'</font> &amp; <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">y</font></span><a href="yellow_3.html#K7" target="_self">]</a></span> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">z'</font> holds <br/><font color="Maroon">z</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">z'</font>
 <b>by </b><i>, <a class="ref" href="lattice3.html#D10">LATTICE3:def 10</a></i>;<br/>
<b>take </b>
<font color="Maroon">z</font> <a href="yellow_3.html#K9" target="_self">`2</a> 
;<br/>

<a NAME="E8:41"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="yellow_3.html#K8" target="_self">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="yellow_3.html#K9" target="_self">`2</a> </span>)</span></span><a href="yellow_3.html#K7" target="_self">]</a></span>
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/>
<b>hence </b><a NAME="E9:41"/>
( <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> <a href="yellow_3.html#K9" target="_self">`2</a>  &amp; <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> <a href="yellow_3.html#K9" target="_self">`2</a>  )
 <b>by </b><i>, </i>;<br/>

<b>let </b><font color="Maroon">z'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font>;<br/>

<b>assume </b><a NAME="E10:41"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z'</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z'</font> )
 ;<br/>

<a NAME="E11:41"/>
<font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font>
 
;<br/>
<a NAME="E12:41"/><b>then </b>
( <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">x</font></span><a href="yellow_3.html#K7" target="_self">]</a></span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">z'</font></span><a href="yellow_3.html#K7" target="_self">]</a></span> &amp; <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">y</font></span><a href="yellow_3.html#K7" target="_self">]</a></span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">z'</font></span><a href="yellow_3.html#K7" target="_self">]</a></span> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E13:41"/><b>then </b>
<font color="Maroon">z</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">z'</font></span><a href="yellow_3.html#K7" target="_self">]</a></span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E14:41"/>
<font color="Maroon">z</font> <a href="yellow_3.html#K9" target="_self">`2</a>  <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z'</font>
 <b>by </b><i>, </i>;<br/>


</div>
