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

<b>set </b><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/>
<b>set </b><font color="Maroon" title="c3">R</font> =  <a href="waybel10.html#K8" title="WAYBEL10:func.8">DsupClOpers</a> <font color="Maroon" title="c1">L</font>;<br/>
<b>set </b><font color="Maroon" title="c4">g</font> =  <a href="waybel_1.html#K2" title="WAYBEL_1:func.2">corestr</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="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <span class="p2">(<span class="default"><a href="waybel10.html#K8" title="WAYBEL10:func.8">DsupClOpers</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:62_1_1_1"/>
( ( not  <a href="waybel10.html#K8" title="WAYBEL10:func.8">DsupClOpers</a> <font color="Maroon" title="c1">L</font> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> &amp; not  <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</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="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <span class="p2">(<span class="default"><a href="waybel10.html#K8" title="WAYBEL10:func.8">DsupClOpers</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> ) or  <a href="waybel10.html#K8" title="WAYBEL10:func.8">DsupClOpers</a> <font color="Maroon" title="c1">L</font> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> or  <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</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="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <span class="p2">(<span class="default"><a href="waybel10.html#K8" title="WAYBEL10:func.8">DsupClOpers</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_1_1"><b>case </b></a><a NAME="E1:62_1_1_1_1"/>
( not  <a href="waybel10.html#K8" title="WAYBEL10:func.8">DsupClOpers</a> <font color="Maroon" title="c1">L</font> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> &amp; not  <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</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="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <span class="p2">(<span class="default"><a href="waybel10.html#K8" title="WAYBEL10:func.8">DsupClOpers</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> )
 ;<br/><div class="add"><a NAME="E2:62_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="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <span class="p1">(<span class="default"><a href="waybel10.html#K8" title="WAYBEL10:func.8">DsupClOpers</a> <font color="Maroon" title="c1">L</font></span>)</span> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="ref" href="waybel10.html#T8" target="_self" title="WAYBEL10:th.8">Th8</a></i>;<br/><b>hence </b><a NAME="E3:62_1_1_1_1"/>
(  <a href="waybel_1.html#K2" title="WAYBEL_1:func.2">corestr</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="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <span class="p2">(<span class="default"><a href="waybel10.html#K8" title="WAYBEL10:func.8">DsupClOpers</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="waybel_1.html#K2" title="WAYBEL_1:func.2">corestr</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="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <span class="p2">(<span class="default"><a href="waybel10.html#K8" title="WAYBEL10:func.8">DsupClOpers</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> )
 <b>by </b><i><a class="ref" href="waybel_1.html#T32" title="WAYBEL_1:th.32">WAYBEL_1:32</a></i>;<br/><b>consider </b><font color="Maroon" title="c5">f'</font> being   <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><b> such that </b><br/><a NAME="E5:62_1_1_1_1"/><i><font color="Green" title="E34">A1</font></i>: 
( <font color="Maroon" title="c5">f'</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">f'</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> )
 <b>by </b><i><a class="ref" href="waybel_0.html#D38" title="WAYBEL_0:def.38">WAYBEL_0:def 38</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">h</font> = <font color="Maroon" title="c5">f'</font> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <span class="p2">(<span class="default"><span class="p3">(<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="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <span class="p3">(<span class="default"><a href="waybel10.html#K8" title="WAYBEL10:func.8">DsupClOpers</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span>)</span> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <span class="p2">(<span class="default"><span class="p3">(<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="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <span class="p3">(<span class="default"><a href="waybel10.html#K8" title="WAYBEL10:func.8">DsupClOpers</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="waybel10.html#K8" title="WAYBEL10:func.8">DsupClOpers</a> <font color="Maroon" title="c1">L</font></span>)</span> <b>by </b><i><a class="txt" href="waybel10.html#E5:62_1_1_1_1"><i><font color="Green" title="E34">A1</font></i></a>, <a class="ref" href="waybel10.html#T9" target="_self" title="WAYBEL10:th.9">Th9</a></i>;<br/><b>take </b>
<font color="Maroon" title="c6">h</font>
;<br/><b>thus </b><font color="Maroon" title="c6">h</font> = 
<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="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <span class="p2">(<span class="default"><a href="waybel10.html#K8" title="WAYBEL10:func.8">DsupClOpers</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> 
<b>by </b><i><a class="txt" href="waybel10.html#E5:62_1_1_1_1"><i><font color="Green" title="E34">A1</font></i></a>, <a class="ref" href="waybel10.html#T9" target="_self" title="WAYBEL10:th.9">Th9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="waybel_1.html#K2" title="WAYBEL_1:func.2">corestr</a> <span class="p2">(<span class="default"><span class="p3">(<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="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <span class="p3">(<span class="default"><a href="waybel10.html#K8" title="WAYBEL10:func.8">DsupClOpers</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> 
<b>by </b><i><a class="ref" href="waybel_1.html#T32" title="WAYBEL_1:th.32">WAYBEL_1:32</a></i>
;<br/><b>let </b><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">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="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <span class="p2">(<span class="default"><span class="p3">(<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="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <span class="p3">(<span class="default"><a href="waybel10.html#K8" title="WAYBEL10:func.8">DsupClOpers</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D2" title="WAYBEL_1:def.2">WAYBEL_1:def 2</a><br/><b>reconsider </b><font color="Maroon" title="c9">a</font> = <font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c10">b</font> = <font color="Maroon" title="c8">y</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> <b>by </b><i><a class="ref" href="yellow_0.html#T59" title="YELLOW_0:th.59">YELLOW_0:59</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">A</font> =  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c9">a</font>, <font color="Maroon" title="c12">B</font> =  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c10">b</font> as  <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> <a href="waybel10.html#NV1" title="WAYBEL10:NV.1">closure</a> <a href="waybel10.html#NM1" title="WAYBEL10:NM.1">System</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="waybel10.html#T17" target="_self" title="WAYBEL10:th.17">Th17</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">i</font> =  <a href="waybel10.html#K7" title="WAYBEL10:func.7">closure_op</a> <font color="Maroon" title="c11">A</font>, <font color="Maroon" title="c14">j</font> =  <a href="waybel10.html#K7" title="WAYBEL10:func.7">closure_op</a> <font color="Maroon" title="c12">B</font> as   <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> <b>by </b><i><a class="ref" href="waybel10.html#T11" target="_self" title="WAYBEL10:th.11">Th11</a></i>;<br/><a NAME="E11:62_1_1_1_1"/>
( <font color="Maroon" title="c5">f'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i</font> &amp; <font color="Maroon" title="c5">f'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j</font> )
 <b>by </b><i><a class="txt" href="waybel10.html#E5:62_1_1_1_1"><i><font color="Green" title="E34">A1</font></i></a>, <a class="ref" href="waybel10.html#T22" target="_self" title="WAYBEL10:th.22">Th22</a></i>;<br/><a NAME="E12:62_1_1_1_1"/><b>then </b><i><font color="Green" title="E35">A2</font></i>: 
( <font color="Maroon" title="c6">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i</font> &amp; <font color="Maroon" title="c6">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j</font> )
 <b>by </b><i><a class="ref" href="waybel10.html#T7" target="_self" title="WAYBEL10:th.7">Th7</a></i>;<br/><b>assume </b><a NAME="E13:62_1_1_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">y</font>
 ;<br/><a NAME="E14:62_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c9">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T60" title="YELLOW_0:th.60">YELLOW_0:60</a></i>;<br/><a NAME="E15:62_1_1_1_1"/><b>then </b>
 <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c9">a</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a>  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="ref" href="yellow_7.html#T1" title="YELLOW_7:th.1">YELLOW_7:1</a></i>;<br/><a NAME="E16:62_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c12">B</font> is    <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c11">A</font> &amp; <font color="Maroon" title="c11">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="c11">A</font></span>)</span> &amp; <font color="Maroon" title="c12">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="c12">B</font></span>)</span> )
 <b>by </b><i><a class="ref" href="waybel10.html#T18" target="_self" title="WAYBEL10:th.18">Th18</a>, <a class="ref" href="waybel10.html#T19" target="_self" title="WAYBEL10:th.19">Th19</a></i>;<br/><a NAME="E17:62_1_1_1_1"/><b>then </b>
 <a href="waybel10.html#K7" title="WAYBEL10:func.7">closure_op</a> <font color="Maroon" title="c11">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="c12">B</font>
 <b>by </b><i><a class="ref" href="waybel10.html#T15" target="_self" title="WAYBEL10:th.15">Th15</a></i>;<br/><a NAME="E18:62_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c13">i</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c14">j</font> &amp; not 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#K8" title="WAYBEL10:func.8">DsupClOpers</a> <font color="Maroon" title="c1">L</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 <b>by </b><i><a class="ref" href="waybel10.html#T13" target="_self" title="WAYBEL10:th.13">Th13</a></i>;<br/><b>hence </b><a NAME="E19:62_1_1_1_1"/>
<font color="Maroon" title="c6">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y</font>
 <b>by </b><i><a class="txt" href="waybel10.html#E12:62_1_1_1_1"><i><font color="Green" title="E35">A2</font></i></a>, <a class="ref" href="yellow_0.html#T61" title="YELLOW_0:th.61">YELLOW_0:61</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_1_2"><b>case </b></a><a NAME="E1:62_1_1_1_2"/>
(  <a href="waybel10.html#K8" title="WAYBEL10:func.8">DsupClOpers</a> <font color="Maroon" title="c1">L</font> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> or  <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</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="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <span class="p2">(<span class="default"><a href="waybel10.html#K8" title="WAYBEL10:func.8">DsupClOpers</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:62_1_1_1_2"/>
(  <a href="waybel10.html#K8" title="WAYBEL10:func.8">DsupClOpers</a> <font color="Maroon" title="c1">L</font> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> &amp;  <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</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="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <span class="p2">(<span class="default"><a href="waybel10.html#K8" title="WAYBEL10:func.8">DsupClOpers</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
