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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V4">antisymmetric</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:70"/><i><font color="Green">E18</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font>
 <b>and </b><br/><a NAME="E2:70"/><i><font color="Green">E19</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>7</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="yellow_3.html#K3" target="_self">:]</a></span>;<br/>

<b>assume </b><a NAME="E3:70"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <span class="p1"><a href="yellow_3.html#K6" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="yellow_3.html#K6" target="_self">:]</a></span>
 ;<br/>

<a NAME="E4:70"/>
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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_3.html#D2" target="_self">Def2</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:70"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></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/>
<a NAME="E7:70"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>7</sub></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">c<sub>7</sub></font> <a href="yellow_3.html#K8" target="_self">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></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="txt" href="yellow_3.html#E6:70"><i><font color="Green">E21</font></i></a>, <a class="ref" href="mcart_1.html#T8">MCART_1:8</a></i>;<br/>
<a NAME="E8:70"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font> <a href="yellow_3.html#K8" target="_self">`1</a>  <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="yellow_3.html#K9" target="_self">`2</a>  <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="yellow_3.html#E3:70"><i><font color="Green">E20</font></i></a>, <a class="ref" href="yellow_3.html#T29" target="_self">Th29</a></i>;<br/>
<a NAME="E9:70"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="yellow_3.html#K8" target="_self">`1</a>  &amp; <font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="yellow_3.html#K9" target="_self">`2</a>  )
 
<b>by </b><i><a class="txt" href="yellow_3.html#E1:70"><i><font color="Green">E18</font></i></a>, <a class="txt" href="yellow_3.html#E2:70"><i><font color="Green">E19</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:70"/>
<span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="yellow_3.html#K7" target="_self">]</a></span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="yellow_3.html#E7:70"><i><font color="Green">E22</font></i></a>, <a class="ref" href="yellow_3.html#T11" target="_self">Th11</a></i>;<br/>


</div>
