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

<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> , <font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</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>, <font color="Maroon">z</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/>



<b>thus </b><a NAME="E1:10"/>
( <font color="Maroon">x</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K7">}</a></span> implies ( <font color="Maroon">x</font> <a href="yellow_3.html#K8">`1</a>  <a href="lattice3.html#R1">is_&lt;=_than</a> <span class="p1"><a href="domain_1.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">z</font> <a href="yellow_3.html#K8">`1</a> </span>)</span></span><a href="domain_1.html#K7">}</a></span> &amp; <font color="Maroon">x</font> <a href="yellow_3.html#K9">`2</a>  <a href="lattice3.html#R1">is_&lt;=_than</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="yellow_3.html#K9">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="yellow_3.html#K9">`2</a> </span>)</span></span><a href="domain_1.html#K7">}</a></span> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:10_1"/><i><font color="Green">E18</font></i>: 
for <font color="Olive">b</font> being  <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>  st <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K7">}</a></span> holds <br/><font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a><br/>

<a NAME="E2:10_1"/><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">[:</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="E3:10_1"/><b>then </b>
( <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> &amp; <font color="Maroon">z</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">z</font> <a href="yellow_3.html#K8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">z</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/>
<a NAME="E4:10_1"/><b>then </b>
( <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> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K7">}</a></span> &amp; <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="yellow_3.html#K8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="yellow_3.html#K9">`2</a> </span>)</span></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K7">}</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E5:10_1"/><b>then </b><i><font color="Green">E21</font></i>: 
( <font color="Maroon">x</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"><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> &amp; <font color="Maroon">x</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"><font color="Maroon">z</font> <a href="yellow_3.html#K8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="yellow_3.html#K9">`2</a> </span>)</span></span><a href="yellow_3.html#K7">]</a></span> )
 
<b>by </b><i/>;<br/>
<a NAME="E6:10_1"/><i><font color="Green">E23</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>
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/>
<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a>
<div class="add"><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/><b>assume </b><a NAME="E1:10_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.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">z</font> <a href="yellow_3.html#K8">`1</a> </span>)</span></span><a href="domain_1.html#K7">}</a></span>
 ;<br/><a NAME="E2:10_1_1"/><b>then </b>
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="yellow_3.html#K8">`1</a>  or <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="yellow_3.html#K8">`1</a>  )
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E3:10_1_1"/>
<font color="Maroon">x</font> <a href="yellow_3.html#K8">`1</a>  <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i>, , <a class="ref" href="yellow_3.html#T11">YELLOW_3:11</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a><br/>

<b>assume </b><a NAME="E8:10_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="yellow_3.html#K9">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="yellow_3.html#K9">`2</a> </span>)</span></span><a href="domain_1.html#K7">}</a></span>
 ;<br/>

<a NAME="E9:10_1"/><b>then </b>
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="yellow_3.html#K9">`2</a>  or <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="yellow_3.html#K9">`2</a>  )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E10:10_1"/>
<font color="Maroon">x</font> <a href="yellow_3.html#K9">`2</a>  <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i>, , <a class="ref" href="yellow_3.html#T11">YELLOW_3:11</a></i>;<br/>


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

<b>assume </b><b>that </b><br/><a NAME="E2:10"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>  st <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.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">z</font> <a href="yellow_3.html#K8">`1</a> </span>)</span></span><a href="domain_1.html#K7">}</a></span> holds <br/><font color="Maroon">x</font> <a href="yellow_3.html#K8">`1</a>  <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b</font>
 <b>and </b><br/><a NAME="E3:10"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>  st <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="yellow_3.html#K9">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="yellow_3.html#K9">`2</a> </span>)</span></span><a href="domain_1.html#K7">}</a></span> holds <br/><font color="Maroon">x</font> <a href="yellow_3.html#K9">`2</a>  <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a><br/>

<b>let </b><font color="Maroon">b</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>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a><br/>

<b>assume </b><a NAME="E4:10"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K7">}</a></span>
 ;<br/>

<a NAME="E5:10"/><b>then </b>
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> or <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E6:10"/><b>then </b>
( <font color="Maroon">b</font> <a href="yellow_3.html#K8">`1</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.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">z</font> <a href="yellow_3.html#K8">`1</a> </span>)</span></span><a href="domain_1.html#K7">}</a></span> &amp; <font color="Maroon">b</font> <a href="yellow_3.html#K9">`2</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="yellow_3.html#K9">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="yellow_3.html#K9">`2</a> </span>)</span></span><a href="domain_1.html#K7">}</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E7:10"/><b>then </b>
( <font color="Maroon">x</font> <a href="yellow_3.html#K8">`1</a>  <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">b</font> <a href="yellow_3.html#K8">`1</a>  &amp; <font color="Maroon">x</font> <a href="yellow_3.html#K9">`2</a>  <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">b</font> <a href="yellow_3.html#K9">`2</a>  )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E8:10"/>
<font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="yellow_3.html#T12">YELLOW_3:12</a></i>;<br/>


</div>
