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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="waybel10.html#K6" target="_self">ClImageMap</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E1:48"/><i><font color="Green">E26</font></i>: 
(  <a href="relat_1.html#K1">dom</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_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</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> &amp;  <a href="relat_1.html#K2">rng</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_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</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> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/>
<a NAME="E2:48"/><i><font color="Green">E27</font></i>: 
(  <a href="relset_1.html#K4">dom</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="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</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> &amp;  <a href="relset_1.html#K5">rng</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="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</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> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:48"/>
the <a href="struct_0.html#U1">carrier</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> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>proof </b></a><div class="add">

<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="E1:48_1"/>
<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"><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>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>4</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"><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> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</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/>
<a NAME="E4:48_1"/><i><font color="Green">E28</font></i>: 
 <a href="waybel10.html#K7" target="_self">closure_op</a> <font color="Maroon">c<sub>5</sub></font> is   <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>
 
<b>by </b><i><a class="ref" href="waybel10.html#T11" target="_self">Th11</a></i>;<br/>
<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#K1">.</a> <span class="p1">(<span class="default"><a href="waybel10.html#K7" target="_self">closure_op</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> = 
 <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>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="waybel10.html#D5" target="_self">Def5</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="waybel10.html#T19" target="_self">Th19</a></i>
;<br/>
<b>hence </b><a NAME="E6:48_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</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>
 <b>by </b><i><a class="txt" href="waybel10.html#E2:48"><i><font color="Green">E27</font></i></a>, <a class="txt" href="waybel10.html#E4:48_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:48"/><b>then </b>
the <a href="struct_0.html#U1">carrier</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> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</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>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E5:48"/>
<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>  is   <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="txt" href="waybel10.html#E1:48"><i><font color="Green">E26</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>


</div>
