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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="lattice3.html#V3">complete</a> <a href="orders_2.html#NM1">Poset</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:95"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="waybel_1.html#NR10" target="_self">is_a_projection_operator</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> , <font color="Maroon">c<sub>4</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="waybel_1.html#T46" target="_self">Th46</a></i>;<br/>
<a NAME="E3:95"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="orders_3.html#V5">monotone</a>
 
<b>by </b><i><a class="txt" href="waybel_1.html#E1:95"><i><font color="Green">E59</font></i></a>, <a class="ref" href="waybel_1.html#D13" target="_self">Def13</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>3</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <b>by </b><i><a class="txt" href="waybel_1.html#E1:95"><i><font color="Green">E59</font></i></a>, <a class="ref" href="waybel_1.html#T48" target="_self">Th48</a></i>;<br/>
<a NAME="E5:95"/>
 <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>3</sub></font> is <a href="yellow_0.html#V8">sups-inheriting</a>
 
<b>by </b><i><a class="txt" href="waybel_1.html#E3:95"><i><font color="Green">E60</font></i></a>, <a class="ref" href="waybel_1.html#T52" target="_self">Th52</a></i>;<br/>
<a NAME="E6:95"/><b>then </b><i><font color="Green">E61</font></i>: 
 <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>3</sub></font> is  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>
 
<b>by </b><i><a class="ref" href="yellow_2.html#T33">YELLOW_2:33</a></i>;<br/>
<a NAME="E7:95"/>
<font color="Maroon">c<sub>5</sub></font> <a href="waybel_1.html#NR11" target="_self">is_a_closure_operator</a> 
 
<b>by </b><i><a class="txt" href="waybel_1.html#E1:95"><i><font color="Green">E59</font></i></a>, <a class="ref" href="waybel_1.html#T50" target="_self">Th50</a></i>;<br/>
<a NAME="E8:95"/><b>then </b>
 <a href="yellow_2.html#K1">Image</a> <font color="Maroon">c<sub>5</sub></font> is <a href="yellow_0.html#V7">infs-inheriting</a>
 
<b>by </b><i><a class="ref" href="waybel_1.html#T56" target="_self">Th56</a></i>;<br/>
<a NAME="E9:95"/><b>then </b>
 <a href="yellow_0.html#K5">subrelstr</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> is <a href="yellow_0.html#V7">infs-inheriting</a>
 
;<br/>
<a NAME="E10:95"/><b>then </b><i><font color="Green">E62</font></i>: 
 <a href="yellow_0.html#K5">subrelstr</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> is <a href="lattice3.html#V3">complete</a>
 
<b>by </b><i><a class="txt" href="waybel_1.html#E6:95"><i><font color="Green">E61</font></i></a>, <a class="ref" href="yellow_2.html#T32">YELLOW_2:32</a></i>;<br/>
<i><font color="Green">E63</font></i>: the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1">Image</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> = 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>

<br/>.= 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>
;<br/>
<a NAME="E12:95"/><i><font color="Green">E64</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<a NAME="E13:95"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_1.html#E1:95"><i><font color="Green">E59</font></i></a>, <a class="ref" href="waybel_1.html#T45" target="_self">Th45</a></i>;<br/>
<a NAME="E14:95"/><b>then </b><i><font color="Green">E65</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1">Image</a> <font color="Maroon">c<sub>2</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"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_1.html#E11:95"><i><font color="Green">E63</font></i></a>, <a class="txt" href="waybel_1.html#E12:95"><i><font color="Green">E64</font></i></a>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<i><font color="Green">E66</font></i>: the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1">Image</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> = 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>

<br/>.= 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1">Image</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="waybel_1.html#E1:95"><i><font color="Green">E59</font></i></a>, <a class="txt" href="waybel_1.html#E11:95"><i><font color="Green">E63</font></i></a>, <a class="ref" href="waybel_1.html#T47" target="_self">Th47</a></i>
;<br/>
<b>then </b>the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1">Image</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> = 
the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="wellord1.html#K2">|_2</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1">Image</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_0.html#D14">YELLOW_0:def 14</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K2">|_2</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="wellord1.html#K2">|_2</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1">Image</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_0.html#D14">YELLOW_0:def 14</a></i>
<br/>.= 
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K2">|_2</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1">Image</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="waybel_1.html#E14:95"><i><font color="Green">E65</font></i></a>, <a class="ref" href="wellord1.html#T29">WELLORD1:29</a></i>
<br/>.= 
the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1">Image</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_0.html#D14">YELLOW_0:def 14</a></i>
;<br/>
<b>hence </b><a NAME="E17:95"/>
 <a href="yellow_2.html#K1">Image</a> <font color="Maroon">c<sub>2</sub></font> is <a href="lattice3.html#V3">complete</a>
 <b>by </b><i><a class="txt" href="waybel_1.html#E10:95"><i><font color="Green">E62</font></i></a>, <a class="txt" href="waybel_1.html#E15:95"><i><font color="Green">E66</font></i></a></i>;<br/>


</div>
