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

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="waybel10.html#K3" target="_self">ClOpers</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">(<span class="default"><a href="waybel10.html#K5" target="_self">ClosureSystems</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="waybel10.html#K1" target="_self">opp</a> ;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="waybel10.html#K6" target="_self">ClImageMap</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">(<span class="default"><a href="waybel10.html#K6" target="_self">ClImageMap</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K2">"</a>  as   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel10.html#K5" target="_self">ClosureSystems</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="waybel10.html#K1" target="_self">opp</a> </span>)</span>,<span class="p1">(<span class="default"><a href="waybel10.html#K3" target="_self">ClOpers</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="waybel10.html#T21" target="_self">Th21</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:50_1_1_1"/>
( ( not  <a href="waybel10.html#K3" target="_self">ClOpers</a> <font color="Maroon">c<sub>1</sub></font> is <a href="struct_0.html#V3">empty</a> &amp; not <span class="p1">(<span class="default"><a href="waybel10.html#K5" target="_self">ClosureSystems</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="waybel10.html#K1" target="_self">opp</a>  is <a href="struct_0.html#V3">empty</a> ) or  <a href="waybel10.html#K3" target="_self">ClOpers</a> <font color="Maroon">c<sub>1</sub></font> is <a href="struct_0.html#V3">empty</a> or <span class="p1">(<span class="default"><a href="waybel10.html#K5" target="_self">ClosureSystems</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="waybel10.html#K1" target="_self">opp</a>  is <a href="struct_0.html#V3">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_1_1"><b>case </b></a><a NAME="E1:50_1_1_1_1"/>
( not  <a href="waybel10.html#K3" target="_self">ClOpers</a> <font color="Maroon">c<sub>1</sub></font> is <a href="struct_0.html#V3">empty</a> &amp; not <span class="p1">(<span class="default"><a href="waybel10.html#K5" target="_self">ClosureSystems</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="waybel10.html#K1" target="_self">opp</a>  is <a href="struct_0.html#V3">empty</a> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:50_1_1_1_1"/>
 <a href="waybel10.html#K6" target="_self">ClImageMap</a> <font color="Maroon">c<sub>1</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 ;<br/><b>thus </b><a NAME="E3:50_1_1_1_1"/>
 <a href="waybel10.html#K6" target="_self">ClImageMap</a> <font color="Maroon">c<sub>1</sub></font> is <a href="orders_3.html#V5">monotone</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel10.html#K3" target="_self">ClOpers</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D2">WAYBEL_1:def 2</a><br/>

<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> as  <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> <b>by </b><i><a class="ref" href="waybel10.html#T11" target="_self">Th11</a></i>;<br/>
<b>assume </b><a NAME="E2:50_1_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<a NAME="E3:50_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="yellow_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="waybel10.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E4:50_1_1_1_1_1"/><b>then </b><i><font color="Green">E27</font></i>: 
 <a href="yellow_2.html#K1">Image</a> <font color="Maroon">c<sub>9</sub></font> is    <a href="yellow_0.html#M1">SubRelStr</a> of  <a href="yellow_2.html#K1">Image</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="waybel10.html#T15" target="_self">Th15</a></i>;<br/>
<a NAME="E5:50_1_1_1_1_1"/><i><font color="Green">E28</font></i>: 
( <span class="p1">(<span class="default"><a href="waybel10.html#K6" target="_self">ClImageMap</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#K1">Image</a> <font color="Maroon">c<sub>8</sub></font> &amp; <span class="p1">(<span class="default"><a href="waybel10.html#K6" target="_self">ClImageMap</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#K1">Image</a> <font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="ref" href="waybel10.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E6:50_1_1_1_1_1"/>
 <a href="lattice3.html#K9">~</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel10.html#K6" target="_self">ClImageMap</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="lattice3.html#K9">~</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel10.html#K6" target="_self">ClImageMap</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel10.html#E4:50_1_1_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="waybel10.html#E5:50_1_1_1_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="waybel10.html#T18" target="_self">Th18</a></i>;<br/>
<b>hence </b><a NAME="E7:50_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="waybel10.html#K6" target="_self">ClImageMap</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><a href="waybel10.html#K6" target="_self">ClImageMap</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="yellow_7.html#T1">YELLOW_7:1</a></i>;<br/>


</div><b>end;</b></div><b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/><b>thus </b><a NAME="E4:50_1_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel10.html#K6" target="_self">ClImageMap</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K2">"</a> 
 ;<br/><b>thus </b><a NAME="E5:50_1_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="orders_3.html#V5">monotone</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel10.html#K5" target="_self">ClosureSystems</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="waybel10.html#K1" target="_self">opp</a> </span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D2">WAYBEL_1:def 2</a><br/>

<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> =  <a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>9</sub></font> =  <a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>7</sub></font> as  <a href="orders_2.html#V1">strict</a> <a href="yellow_0.html#V4">full</a> <a href="yellow_0.html#V7">infs-inheriting</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="waybel10.html#T17" target="_self">Th17</a></i>;<br/>
<b>assume </b><a NAME="E2:50_1_1_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<a NAME="E3:50_1_1_1_1_2"/><b>then </b>
 <a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>7</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_7.html#T1">YELLOW_7:1</a></i>;<br/>
<a NAME="E4:50_1_1_1_1_2"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>9</sub></font> is    <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="waybel10.html#T18" target="_self">Th18</a></i>;<br/>
<a NAME="E5:50_1_1_1_1_2"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#K1">Image</a> <span class="p1">(<span class="default"><a href="waybel10.html#K7" target="_self">closure_op</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#K1">Image</a> <span class="p1">(<span class="default"><a href="waybel10.html#K7" target="_self">closure_op</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="waybel10.html#T19" target="_self">Th19</a></i>;<br/>
<a NAME="E6:50_1_1_1_1_2"/><b>then </b><i><font color="Green">E28</font></i>: 
 <a href="waybel10.html#K7" target="_self">closure_op</a> <font color="Maroon">c<sub>8</sub></font> <a href="yellow_2.html#R1">&lt;=</a>  <a href="waybel10.html#K7" target="_self">closure_op</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="waybel10.html#E4:50_1_1_1_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="waybel10.html#T15" target="_self">Th15</a></i>;<br/>
<a NAME="E7:50_1_1_1_1_2"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel10.html#K7" target="_self">closure_op</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>5</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="waybel10.html#K7" target="_self">closure_op</a> <font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="ref" href="waybel10.html#T22" target="_self">Th22</a></i>;<br/>
<b>thus </b><a NAME="E8:50_1_1_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <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="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="waybel10.html#E6:50_1_1_1_1_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="waybel10.html#E7:50_1_1_1_1_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="waybel10.html#T13" target="_self">Th13</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_1_2"><b>case </b></a><a NAME="E1:50_1_1_1_2"/>
(  <a href="waybel10.html#K3" target="_self">ClOpers</a> <font color="Maroon">c<sub>1</sub></font> is <a href="struct_0.html#V3">empty</a> or <span class="p1">(<span class="default"><a href="waybel10.html#K5" target="_self">ClosureSystems</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="waybel10.html#K1" target="_self">opp</a>  is <a href="struct_0.html#V3">empty</a> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:50_1_1_1_2"/>
(  <a href="waybel10.html#K3" target="_self">ClOpers</a> <font color="Maroon">c<sub>1</sub></font> is <a href="struct_0.html#V3">empty</a> &amp; <span class="p1">(<span class="default"><a href="waybel10.html#K5" target="_self">ClosureSystems</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="waybel10.html#K1" target="_self">opp</a>  is <a href="struct_0.html#V3">empty</a> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
