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

<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>;<br/><b>let </b><font color="Maroon" title="c2">p</font> be   <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>;<br/>



<b>assume </b><a NAME="E1:95"/><i><font color="Green" title="E59">A1</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="waybel_1.html#NR10" title="WAYBEL_1:NR.10">is_a_projection_operator</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c3">Lc</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">c</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font> </span>}</span> , <font color="Maroon" title="c4">Lk</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Maroon" title="c2">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">k</font> </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="waybel_1.html#T46" target="_self" title="WAYBEL_1:th.46">Th46</a></i>;<br/>
<a NAME="E3:95"/><i><font color="Green" title="E60">A2</font></i>: 
<font color="Maroon" title="c2">p</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>
 
<b>by </b><i><a class="txt" href="waybel_1.html#E1:95"><i><font color="Green" title="E59">A1</font></i></a>, <a class="ref" href="waybel_1.html#D13" target="_self" title="WAYBEL_1:def.13">Def13</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">pc</font> = <font color="Maroon" title="c2">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">Lc</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lc</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lc</font></span>)</span> <b>by </b><i><a class="txt" href="waybel_1.html#E1:95"><i><font color="Green" title="E59">A1</font></i></a>, <a class="ref" href="waybel_1.html#T48" target="_self" title="WAYBEL_1:th.48">Th48</a></i>;<br/>
<a NAME="E5:95"/>
 <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lc</font> is <a href="yellow_0.html#V8" title="YELLOW_0:attr.8">sups-inheriting</a>
 
<b>by </b><i><a class="txt" href="waybel_1.html#E3:95"><i><font color="Green" title="E60">A2</font></i></a>, <a class="ref" href="waybel_1.html#T52" target="_self" title="WAYBEL_1:th.52">Th52</a></i>;<br/>
<a NAME="E6:95"/><b>then </b><i><font color="Green" title="E61">A3</font></i>: 
 <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lc</font> is  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>
 
<b>by </b><i><a class="ref" href="yellow_2.html#T33" title="YELLOW_2:th.33">YELLOW_2:33</a></i>;<br/>
<a NAME="E7:95"/>
<font color="Maroon" title="c5">pc</font> <a href="waybel_1.html#NR11" title="WAYBEL_1:NR.11">is_a_closure_operator</a> 
 
<b>by </b><i><a class="txt" href="waybel_1.html#E1:95"><i><font color="Green" title="E59">A1</font></i></a>, <a class="ref" href="waybel_1.html#T50" target="_self" title="WAYBEL_1:th.50">Th50</a></i>;<br/>
<a NAME="E8:95"/><b>then </b>
 <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c5">pc</font> is <a href="yellow_0.html#V7" title="YELLOW_0:attr.7">infs-inheriting</a>
 
<b>by </b><i><a class="ref" href="waybel_1.html#T56" target="_self" title="WAYBEL_1:th.56">Th56</a></i>;<br/>
<a NAME="E9:95"/><b>then </b>
 <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">pc</font></span>)</span> is <a href="yellow_0.html#V7" title="YELLOW_0:attr.7">infs-inheriting</a>
 
;<br/>
<a NAME="E10:95"/><b>then </b><i><font color="Green" title="E62">A4</font></i>: 
 <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">pc</font></span>)</span> is <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>
 
<b>by </b><i><a class="txt" href="waybel_1.html#E6:95"><i><font color="Green" title="E61">A3</font></i></a>, <a class="ref" href="yellow_2.html#T32" title="YELLOW_2:th.32">YELLOW_2:32</a></i>;<br/>
<a NAME="E11:95"/><i><font color="Green" title="E63">A5</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">p</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></i>;<br/>
<a NAME="E12:95"/><i><font color="Green" title="E64">A6</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lc</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Lc</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></i>;<br/>
<a NAME="E13:95"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Lc</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">Lk</font>
 
<b>by </b><i><a class="txt" href="waybel_1.html#E1:95"><i><font color="Green" title="E59">A1</font></i></a>, <a class="ref" href="waybel_1.html#T45" target="_self" title="WAYBEL_1:th.45">Th45</a></i>;<br/>
<a NAME="E14:95"/><b>then </b><i><font color="Green" title="E65">A7</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lc</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_1.html#E11:95"><i><font color="Green" title="E63">A5</font></i></a>, <a class="txt" href="waybel_1.html#E12:95"><i><font color="Green" title="E64">A6</font></i></a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<i><font color="Green" title="E66">A8</font></i>: the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c5">pc</font></span>)</span> = 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">pc</font>
<b>by </b><i><a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></i>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">p</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel_1.html#E1:95"><i><font color="Green" title="E59">A1</font></i></a>, <a class="txt" href="waybel_1.html#E11:95"><i><font color="Green" title="E63">A5</font></i></a>, <a class="ref" href="waybel_1.html#T47" target="_self" title="WAYBEL_1:th.47">Th47</a></i>
;<br/>
<b>then </b>the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c5">pc</font></span>)</span> = 
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lc</font></span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">p</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_0.html#D14" title="YELLOW_0:def.14">YELLOW_0:def 14</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">L</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lc</font></span>)</span></span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">p</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_0.html#D14" title="YELLOW_0:def.14">YELLOW_0:def 14</a></i>
<br/>.= 
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">L</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">p</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel_1.html#E14:95"><i><font color="Green" title="E65">A7</font></i></a>, <a class="ref" href="wellord1.html#T29" title="WELLORD1:th.29">WELLORD1:29</a></i>
<br/>.= 
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">p</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_0.html#D14" title="YELLOW_0:def.14">YELLOW_0:def 14</a></i>
;<br/>
<b>hence </b><a NAME="E17:95"/>
 <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">p</font> is <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>
 <b>by </b><i><a class="txt" href="waybel_1.html#E10:95"><i><font color="Green" title="E62">A4</font></i></a>, <a class="txt" href="waybel_1.html#E15:95"><i><font color="Green" title="E66">A8</font></i></a></i>;<br/>


</div>
