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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  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S1</font>;<br/><b>let </b><font color="Maroon">D2</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S2</font>;<br/>





<b>assume </b><a NAME="E1:76"/><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> )
 ;<br/>

<b>set </b><font color="Maroon">s</font> =  <a href="yellow_0.html#NK3" target="_self">sup</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>;<br/>
<a NAME="E2:76"/>
 <a href="yellow_0.html#NK3" target="_self">sup</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> is    <a href="subset_1.html#M1">Element</a> of <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">S1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">s1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">s2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:76"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">s1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S1</font> &amp; <font color="Maroon">s2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font> &amp;  <a href="yellow_0.html#NK3" target="_self">sup</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> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">s1</font>,<font color="Maroon">s2</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>reconsider </b><font color="Maroon">s1</font> = <font color="Maroon">s1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S1</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">s2</font> = <font color="Maroon">s2</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S2</font> <b>by </b><i/>;<br/>
<a NAME="E7:76"/><i><font color="Green">E25</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</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>,<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>
 
<b>by </b><i>, <a class="txt" href="yellow_3.html#E1:76"><i><font color="Green">E18</font></i></a></i>;<br/>
<a NAME="E8:76"/><b>then </b><i><font color="Green">E28</font></i>: 
<span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">s1</font>,<font color="Maroon">s2</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>, <a class="ref" href="yellow_0.html#T30">YELLOW_0:30</a></i>;<br/>
<a NAME="E9:76"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">s1</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">D1</font>
 
<b>by </b><i/>;<br/>
<a NAME="E10:76"/><i><font color="Green">E30</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">s1</font>,<font color="Maroon">s2</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>
 
<b>by </b><i>, <a class="txt" href="yellow_3.html#E7:76"><i><font color="Green">E25</font></i></a>, <a class="ref" href="yellow_0.html#T30">YELLOW_0:30</a></i>;<br/>
<a NAME="E11:76"/><b>then </b>
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S1</font>  st <font color="Olive">b</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">D1</font> holds <br/><font color="Maroon">s1</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E12:76"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">s1</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</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="E13:76"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">s2</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">D2</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E14:76"/>
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S2</font>  st <font color="Olive">b</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">D2</font> holds <br/><font color="Maroon">s2</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b</font>
 
<b>by </b><i>, <a class="txt" href="yellow_3.html#E8:76"><i><font color="Green">E28</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E15:76"/>
 <a href="yellow_0.html#NK3" target="_self">sup</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> <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"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D1</font></span>)</span>,<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#E9:76"><i><font color="Green">E29</font></i></a>, <a class="txt" href="yellow_3.html#E10:76"><i><font color="Green">E30</font></i></a>, <a class="ref" href="yellow_0.html#T30">YELLOW_0:30</a></i>;<br/>


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