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<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/>
<span class="kw">set </span><font color="Maroon" title="c3">S</font> =  <a href="waybel10.html#K3" title="WAYBEL10:func.3">ClOpers</a> <font color="Maroon" title="c1">L</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">T</font> = <span class="p1">(<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> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">g</font> = <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>  as   <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="ref" href="waybel10.html#T21" target="_self" title="WAYBEL10:th.21">Th21</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:51_1_1_1"/>
( ( not  <a href="waybel10.html#K3" title="WAYBEL10:func.3">ClOpers</a> <font color="Maroon" title="c1">L</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  &amp; not <span class="p1">(<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>  is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  ) or  <a href="waybel10.html#K3" title="WAYBEL10:func.3">ClOpers</a> <font color="Maroon" title="c1">L</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  or <span class="p1">(<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>  is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  )
 </span>;<br/> <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="waybel_0.html#D38" title="WAYBEL_0:def.38">WAYBEL_0:def 38</a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:51_1_1_1_1"/>
( not  <a href="waybel10.html#K3" title="WAYBEL10:func.3">ClOpers</a> <font color="Maroon" title="c1">L</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  &amp; not <span class="p1">(<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>  is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="waybel10.html#K6" title="WAYBEL10:func.6">ClImageMap</a> <font color="Maroon" title="c1">L</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp;  <a href="waybel10.html#K6" title="WAYBEL10:func.6">ClImageMap</a> <font color="Maroon" title="c1">L</font> is  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>  &amp;  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><span class="p3">(<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>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="waybel10.html#K3" title="WAYBEL10:func.3">ClOpers</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  &amp; <font color="Olive">b<sub>1</sub></font> is  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>  ) )</span><br/><div class="add"><span class="kw">thus </span><a NAME="E2:51_1_1_1_1"/>
 <a href="waybel10.html#K6" title="WAYBEL10:func.6">ClImageMap</a> <font color="Maroon" title="c1">L</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="waybel10.html#K6" title="WAYBEL10:func.6">ClImageMap</a> <font color="Maroon" title="c1">L</font> is  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>  &amp;  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><span class="p3">(<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>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="waybel10.html#K3" title="WAYBEL10:func.3">ClOpers</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  &amp; <font color="Olive">b<sub>1</sub></font> is  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>  ) )</span><br/><span class="kw">thus </span><a NAME="E3:51_1_1_1_1"/>
 <a href="waybel10.html#K6" title="WAYBEL10:func.6">ClImageMap</a> <font color="Maroon" title="c1">L</font> is  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><span class="p3">(<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>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="waybel10.html#K3" title="WAYBEL10:func.3">ClOpers</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  &amp; <font color="Olive">b<sub>1</sub></font> is  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font> be   <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="comment"><font color="Red">:: according to </font></span><a class="ref" href="waybel_1.html#D2" title="WAYBEL_1:def.2">WAYBEL_1:def 2</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="c6">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">y</font> or <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_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c8">c</font> = <font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c9">d</font> = <font color="Maroon" title="c7">y</font> as   <a href="waybel_1.html#V7" title="WAYBEL_1:attr.7">closure</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c1">L</font> <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/>
<a NAME="E2:51_1_1_1_1_1"/><span class="lab"><font color="Green" title="E27">A1</font></span>: 
<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_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c9">d</font>
 
<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/>
<span class="kw">assume </span><a NAME="E3:51_1_1_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">y</font>
 ; <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_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y</font></span><br/>

<a NAME="E4:51_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">c</font> <a href="yellow_2.html#R1" title="YELLOW_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel10.html#T13" target="_self" title="WAYBEL10:th.13">Th13</a></span>;<br/>
<a NAME="E5:51_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A2</font></span>: 
 <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c9">d</font> is    <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of  <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c8">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel10.html#T15" target="_self" title="WAYBEL10:th.15">Th15</a></span>;<br/>
<a NAME="E6:51_1_1_1_1_1"/>
<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_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c8">c</font>
 
<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/>
<a NAME="E7:51_1_1_1_1_1"/><span class="kw">then </span>
 <a href="lattice3.html#K9" title="LATTICE3:func.9">~</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_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a>  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</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_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:51_1_1_1_1_1"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="txt" href="#E2:51_1_1_1_1_1"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="waybel10.html#T18" target="_self" title="WAYBEL10:th.18">Th18</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:51_1_1_1_1_1"/>
<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_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_7.html#T1" title="YELLOW_7:th.1">YELLOW_7:1</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><span class="kw">take </span>
<font color="Maroon" title="c5">g</font>
; <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="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  &amp; <font color="Maroon" title="c5">g</font> is  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>  )</span><br/><span class="kw">thus </span><a NAME="E4:51_1_1_1_1"/>
<font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="funct_1.html#K2" title="FUNCT_1:func.2">"</a> 
 ; <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="c5">g</font> is  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> </span><br/><span class="kw">thus </span><a NAME="E5:51_1_1_1_1"/>
<font color="Maroon" title="c5">g</font> is  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font> be   <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>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="waybel_1.html#D2" title="WAYBEL_1:def.2">WAYBEL_1:def 2</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="c6">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">y</font> or <font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c8">A</font> =  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c9">B</font> =  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c7">y</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="E2:51_1_1_1_1_2"/><span class="lab"><font color="Green" title="E27">A3</font></span>: 
<font color="Maroon" title="c9">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c9">B</font></span>)</span>
 
<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/>
<a NAME="E3:51_1_1_1_1_2"/><span class="lab"><font color="Green" title="E28">A4</font></span>: 
<font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel10.html#K7" title="WAYBEL10:func.7">closure_op</a> <font color="Maroon" title="c8">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel10.html#T22" target="_self" title="WAYBEL10:th.22">Th22</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:51_1_1_1_1_2"/>
<font color="Maroon" title="c6">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">y</font>
 ; <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="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y</font></span><br/>

<a NAME="E5:51_1_1_1_1_2"/><span class="kw">then </span>
 <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c7">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c6">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_7.html#T1" title="YELLOW_7:th.1">YELLOW_7:1</a></span>;<br/>
<a NAME="E6:51_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A5</font></span>: 
<font color="Maroon" title="c9">B</font> is    <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c8">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel10.html#T18" target="_self" title="WAYBEL10:th.18">Th18</a></span>;<br/>
<a NAME="E7:51_1_1_1_1_2"/><span class="lab"><font color="Green" title="E30">A6</font></span>: 
<font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel10.html#K7" title="WAYBEL10:func.7">closure_op</a> <font color="Maroon" title="c9">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel10.html#T22" target="_self" title="WAYBEL10:th.22">Th22</a></span>;<br/>
<a NAME="E8:51_1_1_1_1_2"/>
<font color="Maroon" title="c8">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c8">A</font></span>)</span>
 
<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/>
<a NAME="E9:51_1_1_1_1_2"/><span class="kw">then </span>
 <a href="waybel10.html#K7" title="WAYBEL10:func.7">closure_op</a> <font color="Maroon" title="c8">A</font> <a href="yellow_2.html#R1" title="YELLOW_2:pred.1">&lt;=</a>  <a href="waybel10.html#K7" title="WAYBEL10:func.7">closure_op</a> <font color="Maroon" title="c9">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:51_1_1_1_1_2"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="txt" href="#E2:51_1_1_1_1_2"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="ref" href="waybel10.html#T15" target="_self" title="WAYBEL10:th.15">Th15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:51_1_1_1_1_2"/>
<font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1_1_1_1_2"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="txt" href="#E7:51_1_1_1_1_2"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="ref" href="waybel10.html#T13" target="_self" title="WAYBEL10:th.13">Th13</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></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:51_1_1_1_2"/>
(  <a href="waybel10.html#K3" title="WAYBEL10:func.3">ClOpers</a> <font color="Maroon" title="c1">L</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  or <span class="p1">(<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>  is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="waybel10.html#K3" title="WAYBEL10:func.3">ClOpers</a> <font color="Maroon" title="c1">L</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  &amp; <span class="p1">(<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>  is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:51_1_1_1_2"/>
(  <a href="waybel10.html#K3" title="WAYBEL10:func.3">ClOpers</a> <font color="Maroon" title="c1">L</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  &amp; <span class="p1">(<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>  is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  )
 ; <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></div><span class="kw">end;</span></div>

</div>
