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<b>let </b><font color="Maroon">S1</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> , <font color="Maroon">S2</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">D1</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S1</font>;<br/><b>let </b><font color="Maroon">D2</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S2</font>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S1</font>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S2</font>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:68"/><i><font color="Green">E18</font></i>: 
(  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">D1</font>,<font color="Maroon">S1</font> &amp;  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">D2</font>,<font color="Maroon">S2</font> )
 <b>and </b><br/><a NAME="E2:68"/><i><font color="Green">E20</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" target="_self">[:</a><span class="default"><font color="Maroon">S1</font>,<font color="Maroon">S2</font></span><a href="yellow_3.html#K3" target="_self">:]</a></span>  st <font color="Olive">b</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">D1</font>,<font color="Maroon">D2</font></span><a href="yellow_3.html#K6" target="_self">:]</a></span> holds <br/><span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="yellow_3.html#K7" target="_self">]</a></span> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b</font>
 ;<br/>

<b>thus </b><a NAME="E3:68"/>
for <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S1</font>  st <font color="Olive">c</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">D1</font> holds <br/><font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">c</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1"><b>proof </b></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">S1</font>;<br/>

<b>assume </b><a NAME="E1:68_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">b</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">D1</font>
 ;<br/>

<a NAME="E2:68_1"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D2</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">D2</font>
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#T30">YELLOW_0:30</a></i>;<br/>
<a NAME="E3:68_1"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">b</font>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D2</font></span>)</span></span><a href="yellow_3.html#K7" target="_self">]</a></span> <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">D1</font>,<font color="Maroon">D2</font></span><a href="yellow_3.html#K6" target="_self">:]</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:68_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">y</font></span><a href="yellow_3.html#K7" target="_self">]</a></span> <a href="orders_2.html#R1">&lt;=</a> <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">b</font>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D2</font></span>)</span></span><a href="yellow_3.html#K7" target="_self">]</a></span>
 
<b>by </b><i><a class="txt" href="yellow_3.html#E1:68"><i><font color="Green">E18</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:68_1"/>
<font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><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">S2</font>;<br/>

<b>assume </b><a NAME="E4:68"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">b</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">D2</font>
 ;<br/>

<a NAME="E5:68"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D1</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">D1</font>
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#T30">YELLOW_0:30</a></i>;<br/>
<a NAME="E6:68"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D1</font></span>)</span>,<font color="Maroon">b</font></span><a href="yellow_3.html#K7" target="_self">]</a></span> <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">D1</font>,<font color="Maroon">D2</font></span><a href="yellow_3.html#K6" target="_self">:]</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:68"/><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">y</font></span><a href="yellow_3.html#K7" target="_self">]</a></span> <a href="orders_2.html#R1">&lt;=</a> <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D1</font></span>)</span>,<font color="Maroon">b</font></span><a href="yellow_3.html#K7" target="_self">]</a></span>
 
<b>by </b><i><a class="txt" href="yellow_3.html#E1:68"><i><font color="Green">E18</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:68"/>
<font color="Maroon">y</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/>


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