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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="waybel_8.html#V2" target="_self">algebraic</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="waybel_1.html#V7">closure</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:46"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 ;<br/>

<a NAME="E2:46"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is <a href="quantal1.html#V11">idempotent</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="orders_3.html#V5">monotone</a> )
 
<b>by </b><i><a class="ref" href="waybel_1.html#D13">WAYBEL_1:def 13</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E3:46"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="waybel_8.html#K1" target="_self">CompactSublatt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:46"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E6:46"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K1" target="_self">CompactSublatt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E7:46"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="waybel_8.html#E5:46"><i><font color="Green">E24</font></i></a></i>;<br/>
<a NAME="E9:46"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_8.html#E5:46"><i><font color="Green">E24</font></i></a>, <a class="txt" href="waybel_8.html#E7:46"><i><font color="Green">E26</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1">Image</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="waybel_8.html#E9:46"><i><font color="Green">E27</font></i></a></i>;<br/>
<a NAME="E12:46"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="waybel_3.html#V1">compact</a>
 
<b>by </b><i><a class="txt" href="waybel_8.html#E6:46"><i><font color="Green">E25</font></i></a>, <a class="ref" href="waybel_8.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E13:46"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="waybel_3.html#D2">WAYBEL_3:def 2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</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">(<span class="default"><a href="yellow_2.html#K1">Image</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/><a NAME="E1:46_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1">Image</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/><a NAME="E2:46_1"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as  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> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</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_2.html#T7">YELLOW_2:7</a></i>;<br/><b>assume </b><a NAME="E5:46_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><a NAME="E6:46_1"/>
 <a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font> <a href="yellow_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="waybel_1.html#D14">WAYBEL_1:def 14</a></i>;<br/><a NAME="E7:46_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="yellow_2.html#T10">YELLOW_2:10</a></i>;<br/><a NAME="E8:46_1"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="waybel_8.html#E7:46"><i><font color="Green">E26</font></i></a>, <a class="ref" href="tmap_1.html#T91">TMAP_1:91</a></i>;<br/><a NAME="E9:46_1"/><i><font color="Green">E32</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>10</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="E10:46_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#R4">preserves_sup_of</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="waybel_8.html#E1:46"><i><font color="Green">E22</font></i></a>, <a class="ref" href="waybel_0.html#D37">WAYBEL_0:def 37</a></i>;<br/><b>then </b><i><font color="Green">E33</font></i>: <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> = 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="waybel_8.html#E9:46_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a></i>
<br/>.= 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="waybel_8.html#E2:46"><i><font color="Green">E23</font></i></a>, <a class="txt" href="waybel_8.html#E2:46_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="yellow_2.html#T22">YELLOW_2:22</a></i>
;<br/><a NAME="E12:46_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="waybel_8.html#E9:46_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="waybel_1.html#T58">WAYBEL_1:58</a></i>;<br/><a NAME="E13:46_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="waybel_8.html#E5:46_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="waybel_8.html#E11:46_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="yellow_0.html#T60">YELLOW_0:60</a></i>;<br/><a NAME="E14:46_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="waybel_8.html#E8:46_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E16:46_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_8.html#E13:46"><i><font color="Green">E28</font></i></a>, <a class="ref" href="waybel_3.html#D1">WAYBEL_3:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1">Image</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="txt" href="waybel_8.html#E16:46_1"><i><font color="Green">E34</font></i></a></i>;<br/><b>take </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font>;<br/><b>thus </b><a NAME="E18:46_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="waybel_8.html#E16:46_1"><i><font color="Green">E34</font></i></a></i>;<br/><a NAME="E19:46_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1">Image</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/><a NAME="E20:46_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><a NAME="E21:46_1"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="waybel_8.html#E2:46"><i><font color="Green">E23</font></i></a>, <a class="ref" href="yellow_2.html#T21">YELLOW_2:21</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E23:46_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font> )
 ;<br/><a NAME="E24:46_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="waybel_8.html#E2:46"><i><font color="Green">E23</font></i></a>, <a class="txt" href="waybel_8.html#E16:46_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="waybel_1.html#D2">WAYBEL_1:def 2</a></i>;<br/><b>hence </b><a NAME="E25:46_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="waybel_8.html#E7:46"><i><font color="Green">E26</font></i></a>, <a class="txt" href="waybel_8.html#E23:46_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="yellow_0.html#T61">YELLOW_0:61</a></i>;<br/></div><b>end;</b></div>
<a NAME="E15:46"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="waybel_3.html#D1">WAYBEL_3:def 1</a></i>;<br/>
<a NAME="E16:46"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> is <a href="waybel_3.html#V1">compact</a>
 
<b>by </b><i><a class="ref" href="waybel_3.html#D2">WAYBEL_3:def 2</a></i>;<br/>
<a NAME="E17:46"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" target="_self">CompactSublatt</a> <span class="p2">(<span class="default"><a href="yellow_2.html#K1">Image</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel_8.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E18:46"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K1" target="_self">CompactSublatt</a> <span class="p2">(<span class="default"><a href="yellow_2.html#K1">Image</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/>


</div>
