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

<b>let </b><font color="Maroon">S</font> be  <a href="orders_2.html#V4">antisymmetric</a> <a href="lattice3.html#V1">with_suprema</a>  <a href="orders_2.html#L1">RelStr</a> , <font color="Maroon">T</font> be  <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">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">T</font></span><a href="yellow_3.html#K3">:]</a></span>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">T</font></span><a href="yellow_3.html#K3">:]</a></span>;<br/>



<a NAME="E1:16"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">T</font></span><a href="yellow_3.html#K3">:]</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">S</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_3.html#D2">YELLOW_3:def 2</a></i>;<br/>
<a NAME="E2:16"/><b>then </b><i><font color="Green">E18</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="yellow_3.html#K8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="yellow_3.html#K9">`2</a> </span>)</span></span><a href="yellow_3.html#K7">]</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="yellow_3.html#K8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="yellow_3.html#K9">`2</a> </span>)</span></span><a href="yellow_3.html#K7">]</a></span> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/>
<b>set </b><font color="Maroon">a</font> = <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y</font></span>)</span> <a href="yellow_3.html#K8">`1</a> ;<br/>
<b>set </b><font color="Maroon">b</font> = <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y</font></span>)</span> <a href="yellow_3.html#K9">`2</a> ;<br/>
<a NAME="E3:16"/>
( <font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y</font> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/>
<a NAME="E4:16"/><b>then </b><i><font color="Green">E20</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y</font></span>)</span> <a href="yellow_3.html#K8">`1</a>  <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x</font> <a href="yellow_3.html#K8">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y</font></span>)</span> <a href="yellow_3.html#K8">`1</a>  <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y</font> <a href="yellow_3.html#K8">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y</font></span>)</span> <a href="yellow_3.html#K9">`2</a>  <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x</font> <a href="yellow_3.html#K9">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y</font></span>)</span> <a href="yellow_3.html#K9">`2</a>  <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y</font> <a href="yellow_3.html#K9">`2</a>  )
 
<b>by </b><i><a class="ref" href="yellow_3.html#T12">YELLOW_3:12</a></i>;<br/>
<a NAME="E5:16"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">d</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>  st <font color="Olive">d</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x</font> <a href="yellow_3.html#K8">`1</a>  &amp; <font color="Olive">d</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y</font> <a href="yellow_3.html#K8">`1</a>  holds <br/><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y</font></span>)</span> <a href="yellow_3.html#K8">`1</a>  <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:16_1"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">d</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x</font> <a href="yellow_3.html#K8">`1</a>  &amp; <font color="Maroon">d</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y</font> <a href="yellow_3.html#K8">`1</a>  )
 ;<br/>

<b>set </b><font color="Maroon">t</font> = <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="yellow_3.html#K9">`2</a> </span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="yellow_3.html#K9">`2</a> </span>)</span>;<br/>
<a NAME="E2:16_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="yellow_3.html#K9">`2</a> </span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="yellow_3.html#K9">`2</a> </span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x</font> <a href="yellow_3.html#K9">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="yellow_3.html#K9">`2</a> </span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="yellow_3.html#K9">`2</a> </span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y</font> <a href="yellow_3.html#K9">`2</a>  )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/>
<a NAME="E3:16_1"/><b>then </b>
( <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">d</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="yellow_3.html#K9">`2</a> </span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="yellow_3.html#K9">`2</a> </span>)</span></span>)</span></span><a href="yellow_3.html#K7">]</a></span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x</font> &amp; <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">d</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="yellow_3.html#K9">`2</a> </span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="yellow_3.html#K9">`2</a> </span>)</span></span>)</span></span><a href="yellow_3.html#K7">]</a></span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y</font> )
 
<b>by </b><i>, , <a class="ref" href="yellow_3.html#T11">YELLOW_3:11</a></i>;<br/>
<a NAME="E4:16_1"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y</font> <a href="orders_2.html#R1">&lt;=</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">d</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="yellow_3.html#K9">`2</a> </span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="yellow_3.html#K9">`2</a> </span>)</span></span>)</span></span><a href="yellow_3.html#K7">]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/>
<a NAME="E5:16_1"/>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">d</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="yellow_3.html#K9">`2</a> </span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="yellow_3.html#K9">`2</a> </span>)</span></span>)</span></span><a href="yellow_3.html#K7">]</a></span> <a href="yellow_3.html#K8">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">d</font>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<b>hence </b><a NAME="E6:16_1"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y</font></span>)</span> <a href="yellow_3.html#K8">`1</a>  <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">d</font>
 <b>by </b><i>, <a class="ref" href="yellow_3.html#T12">YELLOW_3:12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:16"/>
for <font color="Olive">d</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>  st <font color="Olive">d</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x</font> <a href="yellow_3.html#K9">`2</a>  &amp; <font color="Olive">d</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y</font> <a href="yellow_3.html#K9">`2</a>  holds <br/><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y</font></span>)</span> <a href="yellow_3.html#K9">`2</a>  <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:16_2"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">d</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x</font> <a href="yellow_3.html#K9">`2</a>  &amp; <font color="Maroon">d</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y</font> <a href="yellow_3.html#K9">`2</a>  )
 ;<br/>

<b>set </b><font color="Maroon">s</font> = <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="yellow_3.html#K8">`1</a> </span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="yellow_3.html#K8">`1</a> </span>)</span>;<br/>
<a NAME="E2:16_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="yellow_3.html#K8">`1</a> </span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="yellow_3.html#K8">`1</a> </span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x</font> <a href="yellow_3.html#K8">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="yellow_3.html#K8">`1</a> </span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="yellow_3.html#K8">`1</a> </span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y</font> <a href="yellow_3.html#K8">`1</a>  )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/>
<a NAME="E3:16_2"/><b>then </b>
( <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="yellow_3.html#K8">`1</a> </span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="yellow_3.html#K8">`1</a> </span>)</span></span>)</span>,<font color="Maroon">d</font></span><a href="yellow_3.html#K7">]</a></span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x</font> &amp; <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="yellow_3.html#K8">`1</a> </span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="yellow_3.html#K8">`1</a> </span>)</span></span>)</span>,<font color="Maroon">d</font></span><a href="yellow_3.html#K7">]</a></span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y</font> )
 
<b>by </b><i>, , <a class="ref" href="yellow_3.html#T11">YELLOW_3:11</a></i>;<br/>
<a NAME="E4:16_2"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y</font> <a href="orders_2.html#R1">&lt;=</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="yellow_3.html#K8">`1</a> </span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="yellow_3.html#K8">`1</a> </span>)</span></span>)</span>,<font color="Maroon">d</font></span><a href="yellow_3.html#K7">]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/>
<a NAME="E5:16_2"/>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="yellow_3.html#K8">`1</a> </span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="yellow_3.html#K8">`1</a> </span>)</span></span>)</span>,<font color="Maroon">d</font></span><a href="yellow_3.html#K7">]</a></span> <a href="yellow_3.html#K9">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">d</font>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<b>hence </b><a NAME="E6:16_2"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y</font></span>)</span> <a href="yellow_3.html#K9">`2</a>  <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">d</font>
 <b>by </b><i>, <a class="ref" href="yellow_3.html#T12">YELLOW_3:12</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:16"/>
( <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y</font></span>)</span> <a href="yellow_3.html#K8">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="yellow_3.html#K8">`1</a> </span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="yellow_3.html#K8">`1</a> </span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y</font></span>)</span> <a href="yellow_3.html#K9">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="yellow_3.html#K9">`2</a> </span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="yellow_3.html#K9">`2</a> </span>)</span> )
 <b>by </b><i>, , <a class="ref" href="yellow_0.html#T18">YELLOW_0:18</a></i>;<br/>


</div>
