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

<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="lattice3.html#V3">complete</a> <a href="orders_2.html#NM1">Poset</a>;<br/><b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V2">filtered</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">L</font>,<font color="Maroon">L</font></span><a href="yellow_3.html#K3">:]</a></span>;<br/>



<b>reconsider </b><font color="Maroon">C</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E2:53"/>
<font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> 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">L</font>,<font color="Maroon">L</font></span><a href="yellow_3.html#K3">:]</a></span>
 
;<br/>
<a NAME="E3:53"/><b>then </b>
<font color="Maroon">D</font> <a href="tarski.html#R1">c=</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">L</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</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/>
<b>then reconsider </b><font color="Maroon">D'</font> = <font color="Maroon">D</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">C</font>,<font color="Maroon">C</font></span><a href="zfmisc_1.html#K2">:]</a></span> ;<br/>
<b>set </b><font color="Maroon">D1</font> =  <a href="yellow_3.html#K4">proj1</a> <font color="Maroon">D</font>;<br/>
<b>set </b><font color="Maroon">D2</font> =  <a href="yellow_3.html#K5">proj2</a> <font color="Maroon">D</font>;<br/>
<b>set </b><font color="Maroon">f</font> =  <a href="waybel_2.html#K5" target="_self">sup_op</a> <font color="Maroon">L</font>;<br/>
<a NAME="E5:53"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">D'</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="yellow_3.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_3.html#K4">proj1</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_3.html#K5">proj2</a> <font color="Maroon">D</font></span>)</span></span><a href="yellow_3.html#K6">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_3.html#T1">YELLOW_3:1</a></i>;<br/>
<a NAME="E6:53"/><i><font color="Green">E32</font></i>: 
 <a href="yellow_0.html#R2">ex_inf_of</a> <span class="p1">(<span class="default"><a href="waybel_2.html#K5" target="_self">sup_op</a> <font color="Maroon">L</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D'</font>,<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<a NAME="E7:53"/><i><font color="Green">E34</font></i>: 
 <a href="yellow_0.html#R2">ex_inf_of</a> <span class="p1">(<span class="default"><a href="yellow_3.html#K4">proj1</a> <font color="Maroon">D</font></span>)</span> <a href="yellow_4.html#K2">"\/"</a> <span class="p1">(<span class="default"><a href="yellow_3.html#K5">proj2</a> <font color="Maroon">D</font></span>)</span>,<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<a NAME="E8:53"/><i><font color="Green">E35</font></i>: 
 <a href="yellow_0.html#R2">ex_inf_of</a>  <a href="waybel_0.html#K4">uparrow</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_2.html#K5" target="_self">sup_op</a> <font color="Maroon">L</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font></span>)</span>,<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<a NAME="E9:53"/>
<span class="p1"><a href="yellow_3.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_3.html#K4">proj1</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_3.html#K5">proj2</a> <font color="Maroon">D</font></span>)</span></span><a href="yellow_3.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="yellow_3.html#T49">YELLOW_3:49</a></i>;<br/>
<a NAME="E10:53"/><b>then </b><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><a href="waybel_2.html#K5" target="_self">sup_op</a> <font color="Maroon">L</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="yellow_3.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_3.html#K4">proj1</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_3.html#K5">proj2</a> <font color="Maroon">D</font></span>)</span></span><a href="yellow_3.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="waybel_2.html#K5" target="_self">sup_op</a> <font color="Maroon">L</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">D</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T156">RELAT_1:156</a></i>;<br/>
<a NAME="E11:53"/><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default"><a href="waybel_2.html#K5" target="_self">sup_op</a> <font color="Maroon">L</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">D</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K4">uparrow</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_2.html#K5" target="_self">sup_op</a> <font color="Maroon">L</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E12:53"/>
<span class="p1">(<span class="default"><a href="waybel_2.html#K5" target="_self">sup_op</a> <font color="Maroon">L</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D'</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="waybel_2.html#K5" target="_self">sup_op</a> <font color="Maroon">L</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="yellow_3.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_3.html#K4">proj1</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_3.html#K5">proj2</a> <font color="Maroon">D</font></span>)</span></span><a href="yellow_3.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="waybel_2.html#E7:53"><i><font color="Green">E34</font></i></a>, <a class="ref" href="relat_1.html#T156">RELAT_1:156</a></i>;<br/>
<a NAME="E13:53"/><b>then </b>
<span class="p1">(<span class="default"><a href="waybel_2.html#K5" target="_self">sup_op</a> <font color="Maroon">L</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D'</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="yellow_3.html#K4">proj1</a> <font color="Maroon">D</font></span>)</span> <a href="yellow_4.html#K2">"\/"</a> <span class="p1">(<span class="default"><a href="yellow_3.html#K5">proj2</a> <font color="Maroon">D</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E14:53"/><b>then </b><i><font color="Green">E58</font></i>: 
 <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_2.html#K5" target="_self">sup_op</a> <font color="Maroon">L</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_3.html#K4">proj1</a> <font color="Maroon">D</font></span>)</span> <a href="yellow_4.html#K2">"\/"</a> <span class="p2">(<span class="default"><a href="yellow_3.html#K5">proj2</a> <font color="Maroon">D</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_2.html#E8:53"><i><font color="Green">E35</font></i></a>, <a class="txt" href="waybel_2.html#E10:53"><i><font color="Green">E36</font></i></a>, <a class="ref" href="yellow_0.html#T35">YELLOW_0:35</a></i>;<br/>
<a NAME="E15:53"/><i><font color="Green">E68</font></i>: 
<span class="p1">(<span class="default"><a href="waybel_2.html#K5" target="_self">sup_op</a> <font color="Maroon">L</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="yellow_3.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_3.html#K4">proj1</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_3.html#K5">proj2</a> <font color="Maroon">D</font></span>)</span></span><a href="yellow_3.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K4">uparrow</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_2.html#K5" target="_self">sup_op</a> <font color="Maroon">L</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel_2.html#T1" target="_self">Th1</a>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E16:53"/>
<span class="p1">(<span class="default"><a href="waybel_2.html#K5" target="_self">sup_op</a> <font color="Maroon">L</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="yellow_3.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_3.html#K4">proj1</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_3.html#K5">proj2</a> <font color="Maroon">D</font></span>)</span></span><a href="yellow_3.html#K6">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="yellow_3.html#K4">proj1</a> <font color="Maroon">D</font></span>)</span> <a href="yellow_4.html#K2">"\/"</a> <span class="p1">(<span class="default"><a href="yellow_3.html#K5">proj2</a> <font color="Maroon">D</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E17:53"/><b>then </b>
 <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_3.html#K4">proj1</a> <font color="Maroon">D</font></span>)</span> <a href="yellow_4.html#K2">"\/"</a> <span class="p2">(<span class="default"><a href="yellow_3.html#K5">proj2</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4">uparrow</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_2.html#K5" target="_self">sup_op</a> <font color="Maroon">L</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_2.html#E10:53"><i><font color="Green">E36</font></i></a>, , <a class="ref" href="waybel_2.html#T3" target="_self">Th3</a>, <a class="ref" href="yellow_0.html#T35">YELLOW_0:35</a></i>;<br/>
<a NAME="E18:53"/><b>then </b>
 <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_3.html#K4">proj1</a> <font color="Maroon">D</font></span>)</span> <a href="yellow_4.html#K2">"\/"</a> <span class="p2">(<span class="default"><a href="yellow_3.html#K5">proj2</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_2.html#K5" target="_self">sup_op</a> <font color="Maroon">L</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_2.html#E8:53"><i><font color="Green">E35</font></i></a>, <a class="ref" href="waybel_0.html#T38">WAYBEL_0:38</a></i>;<br/>
<b>hence </b><a NAME="E19:53"/>
 <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_2.html#K5" target="_self">sup_op</a> <font color="Maroon">L</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_3.html#K4">proj1</a> <font color="Maroon">D</font></span>)</span> <a href="yellow_4.html#K2">"\/"</a> <span class="p2">(<span class="default"><a href="yellow_3.html#K5">proj2</a> <font color="Maroon">D</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="orders_2.html#T25">ORDERS_2:25</a></i>;<br/>


</div>
