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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="yellow_0.html#V1">lower-bounded</a> <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="E1:47"/><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="E2:47"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="quantal1.html#V11">idempotent</a>
 
<b>by </b><i><a class="ref" href="waybel_1.html#D13">WAYBEL_1:def 13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> =  <a href="yellow_2.html#K1">Image</a> <font color="Maroon">c<sub>2</sub></font> as  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a> <b>by </b><i><a class="txt" href="waybel_8.html#E1:47"><i><font color="Green">E22</font></i></a>, <a class="ref" href="yellow_2.html#T37">YELLOW_2:37</a></i>;<br/>
<div><i><font color="Green">E23</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:47_1_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E2:47_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E3:47_1_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <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#E1:47_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="waybel_8.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E4:47_1_1"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> is <a href="waybel_3.html#V1">compact</a> )
 <b>by </b><i><a class="txt" href="waybel_8.html#E2:47_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="waybel_8.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E5:47_1_1"/><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> &amp; <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="txt" href="waybel_8.html#E3:47_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="waybel_3.html#D2">WAYBEL_3:def 2</a></i>;<br/><b>take </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>6</sub></font>;<br/><a NAME="E6:47_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="waybel_8.html#E3:47_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="waybel_8.html#E4:47_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/><a NAME="E7:47_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/><a NAME="E8:47_1_1"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_8.html#E5:47_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="waybel_3.html#T2">WAYBEL_3:2</a></i>;<br/><a NAME="E9:47_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="waybel_3.html#T3">WAYBEL_3:3</a></i>;<br/><a NAME="E10:47_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</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/><b>hence </b><a NAME="E11:47_1_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</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#E6:47_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:47_1"/>
( not  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>4</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>4</sub></font> is <a href="waybel_0.html#V1">directed</a> )
 <b>by </b><i><a class="ref" href="waybel_0.html#D1">WAYBEL_0:def 1</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font>;<br/><b>consider </b><font color="Maroon">c<sub>5</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="E2:47_2"/><i><font color="Green">E24</font></i>: 
<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="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="yellow_2.html#T12">YELLOW_2:12</a></i>;<br/><a NAME="E3:47_2"/><i><font color="Green">E25</font></i>: 
( not  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>5</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>5</sub></font> is <a href="waybel_0.html#V1">directed</a> )
 <b>by </b><i><a class="ref" href="waybel_8.html#D4" target="_self">Def4</a></i>;<br/><a NAME="E4:47_2"/><b>then </b><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#R4">preserves_sup_of</a>  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="waybel_8.html#E1:47"><i><font color="Green">E22</font></i></a>, <a class="ref" href="waybel_0.html#D37">WAYBEL_0:def 37</a></i>;<br/><a NAME="E5:47_2"/><i><font color="Green">E27</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="waybel_8.html#E3:47_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="waybel_0.html#T75">WAYBEL_0:75</a></i>;<br/><a NAME="E6:47_2"/>
 <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K8">/\</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> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_8.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E7:47_2"/><b>then </b>
 <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K9">/\</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>
 <b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/><a NAME="E8:47_2"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p3">(<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></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T154">RELAT_1:154</a></i>;<br/><a NAME="E9:47_2"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="orders_3.html#V5">monotone</a>
 <b>by </b><i><a class="txt" href="waybel_8.html#E1:47"><i><font color="Green">E22</font></i></a>, <a class="ref" href="yellow_2.html#T18">YELLOW_2:18</a></i>;<br/><a NAME="E10:47_2"/>
 <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>4</sub></font> is <a href="waybel_0.html#V1">directed</a>
 <b>by </b><i><a class="txt" href="waybel_8.html#E4:47"><i><font color="Green">E23</font></i></a></i>;<br/><a NAME="E11:47_2"/><b>then </b><i><font color="Green">E30</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T75">WAYBEL_0:75</a></i>;<br/><a NAME="E12:47_2"/><i><font color="Green">E31</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/><a NAME="E13:47_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <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="relat_1.html#T144">RELAT_1:144</a></i>;<br/><a NAME="E14:47_2"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><a NAME="E15:47_2"/><i><font color="Green">E33</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/><a NAME="E16:47_2"/><i><font color="Green">E34</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> <span class="p2">(<span class="default"><a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <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> <span class="p2">(<span class="default"><a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_8.html#E4:47_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="waybel_8.html#E5:47_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a></i>;<br/><a NAME="E17:47_2"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E18:47_2"/><b>then </b>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E19:47_2"/><b>then </b>
 <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> <span class="p2">(<span class="default"><a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <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#E16:47_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E20:47_2"/><b>then </b>
 <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> <span class="p2">(<span class="default"><a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <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>
 <b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><a NAME="E21:47_2"/><b>then </b><i><font color="Green">E35</font></i>: 
 <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> <span class="p2">(<span class="default"><a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="waybel_8.html#E14:47_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="waybel_8.html#E15:47_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="yellow_0.html#T65">YELLOW_0:65</a></i>;<br/><i><font color="Green">E36</font></i>: <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> = 
<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> <span class="p2">(<span class="default"><a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="waybel_8.html#D3" target="_self">Def3</a></i>
<br/>.= 
 <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> <span class="p2">(<span class="default"><a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="waybel_8.html#E4:47_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="waybel_8.html#E5:47_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a></i>
;<br/><a NAME="E23:47_2"/><i><font color="Green">E37</font></i>: 
<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> <a href="tarski.html#R1">c=</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="txt" href="waybel_8.html#E1:47"><i><font color="Green">E22</font></i></a>, <a class="ref" href="waybel_8.html#T25" target="_self">Th25</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:47_2_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>6</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="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:47_2_2"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>7</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> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>6</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>7</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>8</sub></font> = <font color="Maroon">c<sub>7</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#E3:47_2_2"><i><font color="Green">E39</font></i></a></i>;<br/><a NAME="E5:47_2_2"/>
<font color="Maroon">c<sub>6</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#E3:47_2_2"><i><font color="Green">E39</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>9</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>3</sub></font> <b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><a NAME="E7:47_2_2"/>
<font color="Maroon">c<sub>6</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>
 <b>by </b><i><a class="txt" href="waybel_8.html#E8:47_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="waybel_8.html#E1:47_2_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E8:47_2_2"/><b>then </b>
<font color="Maroon">c<sub>6</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="txt" href="waybel_8.html#E23:47_2"><i><font color="Green">E37</font></i></a></i>;<br/><a NAME="E9:47_2_2"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>9</sub></font> is <a href="waybel_3.html#V1">compact</a>
 <b>by </b><i><a class="ref" href="waybel_8.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E10:47_2_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="waybel_8.html#E3:47_2_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="waybel_8.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E11:47_2_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</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="txt" href="waybel_8.html#E9:47_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="waybel_1.html#D2">WAYBEL_1:def 2</a></i>;<br/><a NAME="E12:47_2_2"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="waybel_8.html#E2:47_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="waybel_8.html#E3:47_2_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="yellow_0.html#T61">YELLOW_0:61</a></i>;<br/><b>hence </b><a NAME="E13:47_2_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="waybel_8.html#E9:47_2_2"><i><font color="Green">E40</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E25:47_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E26:47_2"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <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_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_8.html#E2:47_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="waybel_8.html#E11:47_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="waybel_8.html#E12:47_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="waybel_8.html#E21:47_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="waybel_8.html#E22:47_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="yellow_0.html#T34">YELLOW_0:34</a></i>;<br/><a NAME="E27:47_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="waybel_8.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E28:47_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a>  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/><a NAME="E29:47_2"/><b>then </b>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T32">YELLOW_0:32</a></i>;<br/><b>hence </b><a NAME="E30:47_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_8.html#E26:47_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="orders_2.html#T25">ORDERS_2:25</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:47"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> is <a href="waybel_8.html#V1" target="_self">satisfying_axiom_K</a>
 
<b>by </b><i><a class="ref" href="waybel_8.html#D3" target="_self">Def3</a></i>;<br/>
<b>hence </b><a NAME="E7:47"/>
 <a href="yellow_2.html#K1">Image</a> <font color="Maroon">c<sub>2</sub></font> is  <a href="waybel_8.html#V2" target="_self">algebraic</a> <a href="waybel_0.html#NM8">LATTICE</a>
 <b>by </b><i><a class="txt" href="waybel_8.html#E4:47"><i><font color="Green">E23</font></i></a>, <a class="ref" href="waybel_8.html#D4" target="_self">Def4</a></i>;<br/>


</div>
