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<b>set </b><font color="Maroon">IT</font> = <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>;<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" 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>, <font color="Maroon">y</font> be   <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>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D11">LATTICE3:def 11</a><br/>

<b>consider </b><font color="Maroon">zx</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E2:36_1_1"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">x</font> <a href="yellow_3.html#K8" target="_self">`1</a>  <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">zx</font> &amp; <font color="Maroon">y</font> <a href="yellow_3.html#K8" target="_self">`1</a>  <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">zx</font> )
 <b>and </b><br/><a NAME="E3:36_1_1"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">z'</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>  st <font color="Maroon">x</font> <a href="yellow_3.html#K8" target="_self">`1</a>  <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">z'</font> &amp; <font color="Maroon">y</font> <a href="yellow_3.html#K8" target="_self">`1</a>  <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">z'</font> holds <br/><font color="Maroon">zx</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">z'</font>
 <b>by </b><i><a class="ref" href="lattice3.html#D11">LATTICE3:def 11</a></i>;<br/>
<b>consider </b><font color="Maroon">zy</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E5:36_1_1"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">x</font> <a href="yellow_3.html#K9" target="_self">`2</a>  <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">zy</font> &amp; <font color="Maroon">y</font> <a href="yellow_3.html#K9" target="_self">`2</a>  <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">zy</font> )
 <b>and </b><br/><a NAME="E6:36_1_1"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">z'</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font>  st <font color="Maroon">x</font> <a href="yellow_3.html#K9" target="_self">`2</a>  <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">z'</font> &amp; <font color="Maroon">y</font> <a href="yellow_3.html#K9" target="_self">`2</a>  <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">z'</font> holds <br/><font color="Maroon">zy</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">z'</font>
 <b>by </b><i><a class="ref" href="lattice3.html#D11">LATTICE3:def 11</a></i>;<br/>
<a NAME="E7:36_1_1"/><i><font color="Green">E29</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>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:36_1_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">c</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">d</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:36_1_1"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> &amp; <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>take </b><font color="Maroon">z</font> = <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">zx</font>,<font color="Maroon">zy</font></span><a href="yellow_3.html#K7" target="_self">]</a></span>;<br/>

<a NAME="E12:36_1_1"/>
( <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="yellow_3.html#K8" target="_self">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="yellow_3.html#K9" target="_self">`2</a> </span>)</span></span><a href="yellow_3.html#K7" target="_self">]</a></span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">zx</font>,<font color="Maroon">zy</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"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="yellow_3.html#K8" target="_self">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="yellow_3.html#K9" target="_self">`2</a> </span>)</span></span><a href="yellow_3.html#K7" target="_self">]</a></span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">zx</font>,<font color="Maroon">zy</font></span><a href="yellow_3.html#K7" target="_self">]</a></span> )
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E13:36_1_1"/>
( <font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">z</font> )
 <b>by </b><i>, , <a class="ref" href="mcart_1.html#T8">MCART_1:8</a></i>;<br/>

<b>let </b><font color="Maroon">z'</font> be   <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>;<br/>

<b>consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E15:36_1_1"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> &amp; <font color="Maroon">z'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>assume </b><a NAME="E16:36_1_1"/>
( <font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">z'</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">z'</font> )
 ;<br/>

<a NAME="E17:36_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="yellow_3.html#K8" target="_self">`1</a>  <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">z'</font> <a href="yellow_3.html#K8" target="_self">`1</a>  &amp; <font color="Maroon">x</font> <a href="yellow_3.html#K9" target="_self">`2</a>  <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">z'</font> <a href="yellow_3.html#K9" target="_self">`2</a>  &amp; <font color="Maroon">y</font> <a href="yellow_3.html#K8" target="_self">`1</a>  <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">z'</font> <a href="yellow_3.html#K8" target="_self">`1</a>  &amp; <font color="Maroon">y</font> <a href="yellow_3.html#K9" target="_self">`2</a>  <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">z'</font> <a href="yellow_3.html#K9" target="_self">`2</a>  )
 
<b>by </b><i/>;<br/>
<a NAME="E18:36_1_1"/><b>then </b>
( <font color="Maroon">zx</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">z'</font> <a href="yellow_3.html#K8" target="_self">`1</a>  &amp; <font color="Maroon">zy</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">z'</font> <a href="yellow_3.html#K9" target="_self">`2</a>  )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E19:36_1_1"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">zx</font>,<font color="Maroon">zy</font></span><a href="yellow_3.html#K7" target="_self">]</a></span> <a href="orders_2.html#NR2" target="_self">&gt;=</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/>;<br/>
<b>hence </b><a NAME="E20:36_1_1"/>
<font color="Maroon">z</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">z'</font>
 <b>by </b><i>, <a class="ref" href="mcart_1.html#T8">MCART_1:8</a></i>;<br/>


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