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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="waybel_0.html#V24">up-complete</a> <a href="waybel_0.html#NM6">Semilattice</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</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">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="yellow_3.html#K3">:]</a></span>;<br/>



<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E2:48"/>
<font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="yellow_3.html#K3">:]</a></span>
 
;<br/>
<a NAME="E3:48"/><b>then </b>
<font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></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">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></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<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> =  <a href="yellow_3.html#K4">proj1</a> <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>6</sub></font> =  <a href="yellow_3.html#K5">proj2</a> <font color="Maroon">c<sub>2</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="yellow_3.html#T21">YELLOW_3:21</a>, <a class="ref" href="yellow_3.html#T22">YELLOW_3:22</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="waybel_2.html#K4" target="_self">inf_op</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E6:48"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="yellow_3.html#K10">[:</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#K10">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_3.html#T1">YELLOW_3:1</a></i>;<br/>
<a NAME="E7:48"/><i><font color="Green">E32</font></i>: 
<span class="p1">(<span class="default"><a href="waybel_2.html#K4" target="_self">inf_op</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>2</sub></font> is <a href="waybel_0.html#V1">directed</a>
 
<b>by </b><i><a class="ref" href="yellow_2.html#T17">YELLOW_2:17</a></i>;<br/>
<a NAME="E8:48"/><b>then </b><i><font color="Green">E33</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1">(<span class="default"><a href="waybel_2.html#K4" target="_self">inf_op</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T75">WAYBEL_0:75</a></i>;<br/>
<a NAME="E9:48"/><i><font color="Green">E34</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>5</sub></font> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T75">WAYBEL_0:75</a></i>;<br/>
<a NAME="E10:48"/>
 <a href="waybel_0.html#K3">downarrow</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_2.html#K4" target="_self">inf_op</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> is <a href="waybel_0.html#V1">directed</a>
 
<b>by </b><i><a class="txt" href="waybel_2.html#E7:48"><i><font color="Green">E32</font></i></a>, <a class="ref" href="waybel_0.html#T30">WAYBEL_0:30</a></i>;<br/>
<a NAME="E11:48"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="waybel_0.html#K3">downarrow</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_2.html#K4" target="_self">inf_op</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T75">WAYBEL_0:75</a></i>;<br/>
<a NAME="E12:48"/>
<span class="p1"><a href="yellow_3.html#K10">[:</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#K10">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_3.html#T48">YELLOW_3:48</a></i>;<br/>
<a NAME="E13:48"/><b>then </b><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><a href="waybel_2.html#K4" target="_self">inf_op</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="yellow_3.html#K10">[:</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#K10">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="waybel_2.html#K4" target="_self">inf_op</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T156">RELAT_1:156</a></i>;<br/>
<a NAME="E14:48"/><i><font color="Green">E37</font></i>: 
<span class="p1">(<span class="default"><a href="waybel_2.html#K4" target="_self">inf_op</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K3">downarrow</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_2.html#K4" target="_self">inf_op</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel_2.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E15:48"/>
<span class="p1">(<span class="default"><a href="waybel_2.html#K4" target="_self">inf_op</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="waybel_2.html#K4" target="_self">inf_op</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="yellow_3.html#K10">[:</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#K10">:]</a></span>
 
<b>by </b><i><a class="txt" href="waybel_2.html#E6:48"><i><font color="Green">E31</font></i></a>, <a class="ref" href="relat_1.html#T156">RELAT_1:156</a></i>;<br/>
<a NAME="E16:48"/><b>then </b>
<span class="p1">(<span class="default"><a href="waybel_2.html#K4" target="_self">inf_op</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="waybel_2.html#T32" target="_self">Th32</a></i>;<br/>
<a NAME="E17:48"/><b>then </b><i><font color="Green">E38</font></i>: 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_2.html#K4" target="_self">inf_op</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_2.html#E8:48"><i><font color="Green">E33</font></i></a>, <a class="txt" href="waybel_2.html#E9:48"><i><font color="Green">E34</font></i></a>, <a class="ref" href="yellow_0.html#T34">YELLOW_0:34</a></i>;<br/>
<a NAME="E18:48"/><i><font color="Green">E39</font></i>: 
<span class="p1">(<span class="default"><a href="waybel_2.html#K4" target="_self">inf_op</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="yellow_3.html#K10">[:</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#K10">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K3">downarrow</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_2.html#K4" target="_self">inf_op</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_2.html#E13:48"><i><font color="Green">E36</font></i></a>, <a class="txt" href="waybel_2.html#E14:48"><i><font color="Green">E37</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E19:48"/>
<span class="p1">(<span class="default"><a href="waybel_2.html#K4" target="_self">inf_op</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="yellow_3.html#K10">[:</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#K10">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="waybel_2.html#T32" target="_self">Th32</a></i>;<br/>
<a NAME="E20:48"/><b>then </b>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K3">downarrow</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_2.html#K4" target="_self">inf_op</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_2.html#E9:48"><i><font color="Green">E34</font></i></a>, <a class="txt" href="waybel_2.html#E11:48"><i><font color="Green">E35</font></i></a>, <a class="txt" href="waybel_2.html#E18:48"><i><font color="Green">E39</font></i></a>, <a class="ref" href="yellow_0.html#T34">YELLOW_0:34</a></i>;<br/>
<a NAME="E21:48"/><b>then </b>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_2.html#K4" target="_self">inf_op</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_2.html#E8:48"><i><font color="Green">E33</font></i></a>, <a class="ref" href="waybel_0.html#T33">WAYBEL_0:33</a></i>;<br/>
<b>hence </b><a NAME="E22:48"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_2.html#K4" target="_self">inf_op</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_3.html#K4">proj1</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="yellow_4.html#K4">"/\"</a> <span class="p2">(<span class="default"><a href="yellow_3.html#K5">proj2</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_2.html#E17:48"><i><font color="Green">E38</font></i></a>, <a class="ref" href="orders_2.html#T25">ORDERS_2:25</a></i>;<br/>


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