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

<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>  <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="waybel10.html#K6" title="WAYBEL10:func.6">ClImageMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  is   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel10.html#K5" title="WAYBEL10:func.5">ClosureSystems</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="waybel10.html#K1" title="WAYBEL10:func.1">opp</a> </span>)</span>,<span class="p1">(<span class="default"><a href="waybel10.html#K3" title="WAYBEL10:func.3">ClOpers</a> <font color="Maroon" title="c1">L</font></span>)</span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">f</font> =  <a href="waybel10.html#K6" title="WAYBEL10:func.6">ClImageMap</a> <font color="Maroon" title="c1">L</font>;<br/>
<a NAME="E1:49"/><span class="lab"><font color="Green" title="E26">A1</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel10.html#K6" title="WAYBEL10:func.6">ClImageMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="waybel10.html#K6" title="WAYBEL10:func.6">ClImageMap</a> <font color="Maroon" title="c1">L</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></span>;<br/>
<a NAME="E2:49"/><span class="lab"><font color="Green" title="E27">A2</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="waybel10.html#K6" title="WAYBEL10:func.6">ClImageMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel10.html#K3" title="WAYBEL10:func.3">ClOpers</a> <font color="Maroon" title="c1">L</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E3:49"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel10.html#K5" title="WAYBEL10:func.5">ClosureSystems</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="waybel10.html#K1" title="WAYBEL10:func.1">opp</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="waybel10.html#K6" title="WAYBEL10:func.6">ClImageMap</a> <font color="Maroon" title="c1">L</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel10.html#K5" title="WAYBEL10:func.5">ClosureSystems</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="waybel10.html#K1" title="WAYBEL10:func.1">opp</a> </span>)</span> or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="waybel10.html#K6" title="WAYBEL10:func.6">ClImageMap</a> <font color="Maroon" title="c1">L</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:49_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel10.html#K5" title="WAYBEL10:func.5">ClosureSystems</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="waybel10.html#K1" title="WAYBEL10:func.1">opp</a> </span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="waybel10.html#K6" title="WAYBEL10:func.6">ClImageMap</a> <font color="Maroon" title="c1">L</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">x</font> = <font color="Maroon" title="c3">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel10.html#K5" title="WAYBEL10:func.5">ClosureSystems</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="waybel10.html#K1" title="WAYBEL10:func.1">opp</a> </span>)</span> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c4">x</font> as   <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>   <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>   <a href="yellow_0.html#V7" title="YELLOW_0:attr.7">infs-inheriting</a>   <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel10.html#T17" target="_self" title="WAYBEL10:th.17">Th17</a></span>;<br/>
<a NAME="E4:49_1"/><span class="lab"><font color="Green" title="E28">A3</font></span>: 
 <a href="waybel10.html#K7" title="WAYBEL10:func.7">closure_op</a> <font color="Maroon" title="c5">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel10.html#K3" title="WAYBEL10:func.3">ClOpers</a> <font color="Maroon" title="c1">L</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel10.html#T11" target="_self" title="WAYBEL10:th.11">Th11</a></span>;<br/>
<span class="p1">(<span class="default"><a href="waybel10.html#K6" title="WAYBEL10:func.6">ClImageMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="waybel10.html#K7" title="WAYBEL10:func.7">closure_op</a> <font color="Maroon" title="c5">x</font></span>)</span> = 
 <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <span class="p1">(<span class="default"><a href="waybel10.html#K7" title="WAYBEL10:func.7">closure_op</a> <font color="Maroon" title="c5">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel10.html#D5" target="_self" title="WAYBEL10:def.5">Def5</a></span>
<br/>.= 
<font color="Maroon" title="c5">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel10.html#T19" target="_self" title="WAYBEL10:th.19">Th19</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:49_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="waybel10.html#K6" title="WAYBEL10:func.6">ClImageMap</a> <font color="Maroon" title="c1">L</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:49"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="txt" href="#E4:49_1"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:49"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A4</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel10.html#K5" title="WAYBEL10:func.5">ClosureSystems</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="waybel10.html#K1" title="WAYBEL10:func.1">opp</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="waybel10.html#K6" title="WAYBEL10:func.6">ClImageMap</a> <font color="Maroon" title="c1">L</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E5:49"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel10.html#K6" title="WAYBEL10:func.6">ClImageMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="waybel10.html#K6" title="WAYBEL10:func.6">ClImageMap</a> <font color="Maroon" title="c1">L</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:49"/>
<span class="p1">(<span class="default"><a href="waybel10.html#K6" title="WAYBEL10:func.6">ClImageMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  is   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel10.html#K5" title="WAYBEL10:func.5">ClosureSystems</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="waybel10.html#K1" title="WAYBEL10:func.1">opp</a> </span>)</span>,<span class="p1">(<span class="default"><a href="waybel10.html#K3" title="WAYBEL10:func.3">ClOpers</a> <font color="Maroon" title="c1">L</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E4:49"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
