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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="yellow_1.html#V1">RelStr-yielding</a> <a href="waybel_3.html#V4">non-Empty</a> <a href="waybel_3.html#V5">reflexive-yielding</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:28"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6">.</a> <font color="Olive">b<sub>1</sub></font> is  <a href="yellow_0.html#V1">lower-bounded</a> <a href="waybel_8.html#V2">algebraic</a> <a href="waybel_0.html#NM8">LATTICE</a>
 ;<br/>

<a NAME="E2:28"/><b>then </b><i><font color="Green">E24</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6">.</a> <font color="Olive">b<sub>1</sub></font> is  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>
 
;<br/>
<a NAME="E3:28"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6">.</a> <font color="Olive">b<sub>1</sub></font> is <a href="orders_2.html#V4">antisymmetric</a>
 
<b>by </b><i><a class="txt" href="waybel13.html#E1:28"><i><font color="Green">E23</font></i></a></i>;<br/>
<a NAME="E4:28"/><b>then </b><i><font color="Green">E25</font></i>: 
 <a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font> is <a href="orders_2.html#V4">antisymmetric</a>
 
<b>by </b><i><a class="ref" href="waybel_3.html#T30">WAYBEL_3:30</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> =  <a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font> as  non <a href="struct_0.html#V3">empty</a> <a href="yellow_0.html#V1">lower-bounded</a> <a href="waybel_0.html#NM8">LATTICE</a> <b>by </b><i><a class="txt" href="waybel13.html#E2:28"><i><font color="Green">E24</font></i></a>, <a class="ref" href="waybel_3.html#T31">WAYBEL_3:31</a></i>;<br/>
<a NAME="E6:28"/><i><font color="Green">E26</font></i>: 
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> holds <br/>( not  <a href="waybel_8.html#K2">compactbelow</a> <font color="Olive">b<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="waybel_8.html#K2">compactbelow</a> <font color="Olive">b<sub>1</sub></font> is <a href="waybel_0.html#V1">directed</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>proof </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="28_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><a NAME="E1:28_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_8.html#K2">compactbelow</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_8.html#K2">compactbelow</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/><a NAME="E2:28_1_1"/><b>then </b><i><font color="Green">E27</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> &amp; <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="ref" href="waybel_8.html#T4">WAYBEL_8:4</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="E3:28_1_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/><a NAME="E4:28_1_1"/>
<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="txt" href="waybel13.html#E2:28_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="waybel_3.html#D2">WAYBEL_3:def 2</a></i>;<br/><a NAME="E5:28_1_1"/><b>then </b><i><font color="Green">E29</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> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="waybel13.html#E3:28_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="E6:28_1_1"/>
<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="waybel13.html#E2:28_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="waybel_3.html#D2">WAYBEL_3:def 2</a></i>;<br/><a NAME="E7:28_1_1"/><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>5</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="waybel13.html#E3:28_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="E8:28_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>6</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">c<sub>5</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="waybel13.html#E5:28_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="waybel_3.html#T3">WAYBEL_3:3</a></i>;<br/><a NAME="E9:28_1_1"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="lattice3.html#K13">"\/"</a> <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="E10:28_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>6</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="waybel13.html#E2:28_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/><b>hence </b><a NAME="E11:28_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_8.html#K2">compactbelow</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="waybel13.html#E9:28_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="waybel_8.html#T4">WAYBEL_8:4</a></i>;<br/><b>thus </b><a NAME="E12:28_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/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:28_1"/>
( not  <a href="waybel_8.html#K2">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">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>
<a NAME="E7:28"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="waybel_0.html#V24">up-complete</a>
 
<b>by </b><i><a class="txt" href="waybel13.html#E2:28"><i><font color="Green">E24</font></i></a>, <a class="ref" href="waybel_3.html#T31">WAYBEL_3:31</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_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 <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/><a NAME="E1:28_2"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2">compactbelow</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="waybel_3.html#K7">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</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">compactbelow</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="waybel_3.html#K7">.</a> <font color="Olive">b<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E1:28_2_1"/><i><font color="Green">E29</font></i>: 
 <a href="waybel_3.html#K5">pi</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2">compactbelow</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="waybel_8.html#K2">compactbelow</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</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="E1:28_2_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K5">pi</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2">compactbelow</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:28_2_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_8.html#K2">compactbelow</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E4:28_2_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="card_3.html#D6">CARD_3:def 6</a></i>;<br/>
<a NAME="E5:28_2_1_1"/>
<font color="Maroon">c<sub>7</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 <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : ( <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="waybel_3.html#V1">compact</a> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="waybel13.html#E3:28_2_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="waybel_8.html#D2">WAYBEL_8:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E7:28_2_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E8:28_2_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E9:28_2_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>8</sub></font> is <a href="waybel_3.html#V1">compact</a>
 ;<br/>
<a NAME="E10:28_2_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="waybel13.html#E9:28_2_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="waybel_3.html#D2">WAYBEL_3:def 2</a></i>;<br/>
<a NAME="E11:28_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">c<sub>8</sub></font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="waybel13.html#E2:28"><i><font color="Green">E24</font></i></a>, <a class="ref" href="waybel_3.html#T33">WAYBEL_3:33</a></i>;<br/>
<a NAME="E12:28_2_1_1"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c<sub>5</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="E13:28_2_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="waybel13.html#E8:28_2_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="waybel_3.html#T28">WAYBEL_3:28</a></i>;<br/>
<b>hence </b><a NAME="E14:28_2_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_8.html#K2">compactbelow</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel13.html#E4:28_2_1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="waybel13.html#E7:28_2_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="waybel13.html#E12:28_2_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="waybel_8.html#T4">WAYBEL_8:4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:28_2_1"/>
 <a href="waybel_8.html#K2">compactbelow</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="waybel_3.html#K5">pi</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2">compactbelow</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</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="E1:28_2_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_8.html#K2">compactbelow</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:28_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>6</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 <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> : ( <font color="Maroon">c<sub>4</sub></font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="waybel_3.html#V1">compact</a> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="waybel_8.html#D2">WAYBEL_8:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span><b> such that </b><br/><a NAME="E4:28_2_1_2"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E5:28_2_1_2"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E6:28_2_1_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>7</sub></font> is <a href="waybel_3.html#V1">compact</a>
 ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> = <font color="Maroon">c<sub>7</sub></font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> =  <a href="yellow_0.html#K3">Bottom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E8:28_2_1_2"/><i><font color="Green">E33</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E9:28_2_1_2"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] implies <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) &amp; (  not <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] implies <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) )
 <b>from </b><i><a class="ref" href="partfun1.html#S6">PARTFUN1:sch 6</a>();<br/></i>
<a NAME="E10:28_2_1_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="waybel13.html#E9:28_2_1_2"><i><font color="Green">E34</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_2_1_2_1_1_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:28_2_1_2_1_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:28_2_1_2_1_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel13.html#E9:28_2_1_2"><i><font color="Green">E34</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:28_2_1_2_1_1_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><a NAME="E2:28_2_1_2_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel13.html#E9:28_2_1_2"><i><font color="Green">E34</font></i></a></i>;<br/><b>hence </b><a NAME="E3:28_2_1_2_1_1_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E2:28_2_1_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="txt" href="waybel13.html#E8:28_2_1_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="waybel_3.html#T27">WAYBEL_3:27</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E1:28_2_1_2_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6">.</a> <font color="Maroon">c<sub>10</sub></font> is  <a href="yellow_0.html#V1">lower-bounded</a> <a href="waybel_8.html#V2">algebraic</a> <a href="waybel_0.html#NM8">LATTICE</a>
 <b>by </b><i><a class="txt" href="waybel13.html#E1:28"><i><font color="Green">E23</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_1_2_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_1_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_2_1_2_2_1_1"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_1_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:28_2_1_2_2_1_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:28_2_1_2_2_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="waybel13.html#E5:28_2_1_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="waybel13.html#E9:28_2_1_2"><i><font color="Green">E34</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_1_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:28_2_1_2_2_1_1_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><a NAME="E2:28_2_1_2_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel13.html#E9:28_2_1_2"><i><font color="Green">E34</font></i></a></i>;<br/><b>hence </b><a NAME="E3:28_2_1_2_2_1_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="waybel13.html#E1:28_2_1_2_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="yellow_0.html#T44">YELLOW_0:44</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:28_2_1_2_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/></div><b>end;</b></div>
<a NAME="E14:28_2_1_2"/><b>then </b><i><font color="Green">E36</font></i>: 
<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="ref" href="waybel_3.html#T28">WAYBEL_3:28</a></i>;<br/>
<a NAME="E15:28_2_1_2"/><i><font color="Green">E37</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>9</sub></font> <a href="waybel_3.html#K7">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_1_2_3"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span>;<br/>

<b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:28_2_1_2_3"/>
not <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<a NAME="E2:28_2_1_2_3"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:28_2_1_2_3"/>
<font color="Maroon">c<sub>9</sub></font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel13.html#E9:28_2_1_2"><i><font color="Green">E34</font></i></a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_1_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E1:28_2_1_2_4"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6">.</a> <font color="Maroon">c<sub>10</sub></font> is  <a href="yellow_0.html#V1">lower-bounded</a> <a href="waybel_8.html#V2">algebraic</a> <a href="waybel_0.html#NM8">LATTICE</a>
 <b>by </b><i><a class="txt" href="waybel13.html#E1:28"><i><font color="Green">E23</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_1_2_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_1_2_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_2_1_2_4_1_1"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_1_2_4_1_1_1"><b>suppose </b></a><a NAME="E1:28_2_1_2_4_1_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><a NAME="E2:28_2_1_2_4_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="waybel13.html#E9:28_2_1_2"><i><font color="Green">E34</font></i></a></i>;<br/><b>hence </b><a NAME="E3:28_2_1_2_4_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">c<sub>9</sub></font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="waybel13.html#E6:28_2_1_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="waybel13.html#E1:28_2_1_2_4_1_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="waybel_3.html#D2">WAYBEL_3:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_1_2_4_1_1_2"><b>suppose </b></a><a NAME="E1:28_2_1_2_4_1_1_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><a NAME="E2:28_2_1_2_4_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel13.html#E9:28_2_1_2"><i><font color="Green">E34</font></i></a></i>;<br/><a NAME="E3:28_2_1_2_4_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c<sub>10</sub></font> is <a href="waybel_3.html#V1">compact</a>
 <b>by </b><i><a class="txt" href="waybel13.html#E1:28_2_1_2_4"><i><font color="Green">E38</font></i></a>, <a class="ref" href="waybel_3.html#T15">WAYBEL_3:15</a></i>;<br/><b>hence </b><a NAME="E4:28_2_1_2_4_1_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">c<sub>9</sub></font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="waybel_3.html#D2">WAYBEL_3:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:28_2_1_2_4"/>
<font color="Maroon">c<sub>9</sub></font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">c<sub>9</sub></font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/></div><b>end;</b></div>
<a NAME="E17:28_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="waybel13.html#E2:28"><i><font color="Green">E24</font></i></a>, <a class="txt" href="waybel13.html#E15:28_2_1_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="waybel_3.html#T33">WAYBEL_3:33</a></i>;<br/>
<a NAME="E18:28_2_1_2"/><b>then </b>
<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_3.html#D2">WAYBEL_3:def 2</a></i>;<br/>
<a NAME="E19:28_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_8.html#K2">compactbelow</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="waybel13.html#E14:28_2_1_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="waybel_8.html#T4">WAYBEL_8:4</a></i>;<br/>
<b>hence </b><a NAME="E20:28_2_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K5">pi</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2">compactbelow</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="waybel13.html#E4:28_2_1_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="waybel13.html#E10:28_2_1_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="card_3.html#D6">CARD_3:def 6</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b> <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2">compactbelow</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> = 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K5">pi</a> <span class="p2">(<span class="default"><a href="waybel_8.html#K2">compactbelow</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="waybel13.html#E1:28_2_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>
<br/>.= 
<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">compactbelow</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="waybel13.html#E2:28"><i><font color="Green">E24</font></i></a>, <a class="ref" href="waybel_3.html#T32">WAYBEL_3:32</a></i>
;<br/><br/>


</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E1:28_2_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6">.</a> <font color="Maroon">c<sub>5</sub></font> is <a href="waybel_8.html#V1">satisfying_axiom_K</a>
 <b>by </b><i><a class="txt" href="waybel13.html#E1:28"><i><font color="Green">E23</font></i></a></i>;<br/><a NAME="E2:28_2_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c<sub>5</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">compactbelow</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="waybel_3.html#K7">.</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">WAYBEL_8:def 3</a></i>;<br/><b>hence </b><a NAME="E3:28_2_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</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">compactbelow</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="waybel13.html#E1:28_2"><i><font color="Green">E28</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E3:28_2"/><b>then </b><i><font color="Green">E29</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">compactbelow</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_3.html#T28">WAYBEL_3:28</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E1:28_2_3"/>
<font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6">.</a> <font color="Maroon">c<sub>5</sub></font> is <a href="waybel_8.html#V1">satisfying_axiom_K</a>
 <b>by </b><i><a class="txt" href="waybel13.html#E1:28"><i><font color="Green">E23</font></i></a></i>;<br/><a NAME="E2:28_2_3"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c<sub>5</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">compactbelow</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="waybel_3.html#K7">.</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">WAYBEL_8:def 3</a></i>;<br/><b>hence </b><a NAME="E3:28_2_3"/>
<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">compactbelow</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="waybel13.html#E1:28_2"><i><font color="Green">E28</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E5:28_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">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="waybel_3.html#T28">WAYBEL_3:28</a></i>;<br/><b>hence </b><a NAME="E6:28_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">compactbelow</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel13.html#E4:28"><i><font color="Green">E25</font></i></a>, <a class="txt" href="waybel13.html#E3:28_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="yellow_0.html#D3">YELLOW_0:def 3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:28"/><b>then </b>
 <a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font> is <a href="waybel_8.html#V1">satisfying_axiom_K</a>
 
<b>by </b><i><a class="ref" href="waybel_8.html#D3">WAYBEL_8:def 3</a></i>;<br/>
<b>hence </b><a NAME="E10:28"/>
 <a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font> is  <a href="yellow_0.html#V1">lower-bounded</a> <a href="waybel_8.html#V2">algebraic</a> <a href="waybel_0.html#NM8">LATTICE</a>
 <b>by </b><i><a class="txt" href="waybel13.html#E6:28"><i><font color="Green">E26</font></i></a>, <a class="txt" href="waybel13.html#E7:28"><i><font color="Green">E27</font></i></a>, <a class="ref" href="waybel_8.html#D4">WAYBEL_8:def 4</a></i>;<br/>


</div>
