<?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="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><b>that </b><br/><a NAME="E1:89"/><i><font color="Green" title="E56">A1</font></i>: 
<font color="Maroon" title="c2">p</font> is <a href="quantal1.html#V11" title="QUANTAL1:attr.11">idempotent</a>
 <b>and </b><br/><a NAME="E2:89"/><i><font color="Green" title="E57">A2</font></i>: 
<font color="Maroon" title="c2">p</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D13" target="_self" title="WAYBEL_1:def.13">WAYBEL_1:def 13</a><br/>

<b>let </b><font color="Maroon" title="c3">Lk</font> be  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>;<br/>

<b>assume </b><a NAME="E3:89"/><i><font color="Green" title="E58">A3</font></i>: 
<font color="Maroon" title="c3">Lk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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> 
 ;<br/>

<b>let </b><font color="Maroon" title="c4">pk</font> be   <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">Lk</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">Lk</font></span>)</span>;<br/>

<b>assume </b><a NAME="E4:89"/><i><font color="Green" title="E59">A4</font></i>: 
<font color="Maroon" title="c4">pk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">Lk</font>
 ;<br/>

<a NAME="E5:89"/><i><font color="Green" title="E60">A5</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">pk</font> <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="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lk</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="quantal1.html#D9" title="QUANTAL1:def.9">QUANTAL1:def 9</a>,<a class="ref" href="waybel_1.html#D13" target="_self" title="WAYBEL_1:def.13">WAYBEL_1:def 13</a>,<a class="ref" href="waybel_1.html#D15" target="_self" title="WAYBEL_1:def.15">WAYBEL_1:def 15</a>
<div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</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_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lk</font></span>)</span>;<br/><a NAME="E1:89_1_1"/><i><font color="Green" title="E61">A6</font></i>: 
<font color="Maroon" title="c4">pk</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="txt" href="waybel_1.html#E4:89"><i><font color="Green" title="E59">A4</font></i></a>, <a class="txt" href="waybel_1.html#E5:89"><i><font color="Green" title="E60">A5</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/><a NAME="E2:89_1_1"/><i><font color="Green" title="E62">A7</font></i>: 
<font color="Maroon" title="c5">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T59" title="YELLOW_0:th.59">YELLOW_0:59</a></i>;<br/><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span> = 
<font color="Maroon" title="c4">pk</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">pk</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel_1.html#E4:89"><i><font color="Green" title="E59">A4</font></i></a>, <a class="txt" href="waybel_1.html#E5:89"><i><font color="Green" title="E60">A5</font></i></a>, <a class="txt" href="waybel_1.html#E1:89_1_1"><i><font color="Green" title="E61">A6</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">pk</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">pk</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font>
<b>by </b><i><a class="txt" href="waybel_1.html#E5:89"><i><font color="Green" title="E60">A5</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
;<br/><b>hence </b><a NAME="E4:89_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">pk</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">pk</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">pk</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="txt" href="waybel_1.html#E1:89"><i><font color="Green" title="E56">A1</font></i></a>, <a class="txt" href="waybel_1.html#E1:89_1_1"><i><font color="Green" title="E61">A6</font></i></a>, <a class="txt" href="waybel_1.html#E2:89_1_1"><i><font color="Green" title="E62">A7</font></i></a>, <a class="ref" href="yellow_2.html#T20" title="YELLOW_2:th.20">YELLOW_2:20</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:89_1"/>
<font color="Maroon" title="c4">pk</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">pk</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">pk</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/><b>thus </b><a NAME="E3:89_1"/>
<font color="Maroon" title="c4">pk</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x1</font>, <font color="Maroon" title="c6">x2</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_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lk</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D2" target="_self" title="WAYBEL_1:def.2">WAYBEL_1:def 2</a><br/>

<b>reconsider </b><font color="Maroon" title="c7">x1'</font> = <font color="Maroon" title="c5">x1</font>, <font color="Maroon" title="c8">x2'</font> = <font color="Maroon" title="c6">x2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="yellow_0.html#T59" title="YELLOW_0:th.59">YELLOW_0:59</a></i>;<br/>
<a NAME="E2:89_1_2"/><i><font color="Green" title="E61">A8</font></i>: 
( <font color="Maroon" title="c4">pk</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x1'</font> &amp; <font color="Maroon" title="c4">pk</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x2'</font> )
 
<b>by </b><i><a class="txt" href="waybel_1.html#E4:89"><i><font color="Green" title="E59">A4</font></i></a>, <a class="txt" href="waybel_1.html#E5:89"><i><font color="Green" title="E60">A5</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>
<b>assume </b><a NAME="E3:89_1_2"/>
<font color="Maroon" title="c5">x1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">x2</font>
 ;<br/>

<a NAME="E4:89_1_2"/><b>then </b>
<font color="Maroon" title="c7">x1'</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">x2'</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="E5:89_1_2"/><b>then </b>
<font color="Maroon" title="c2">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x1'</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="Maroon" title="c8">x2'</font>
 
<b>by </b><i><a class="txt" href="waybel_1.html#E2:89"><i><font color="Green" title="E57">A2</font></i></a>, <a class="ref" href="waybel_1.html#D2" target="_self" title="WAYBEL_1:def.2">Def2</a></i>;<br/>
<b>hence </b><a NAME="E6:89_1_2"/>
<font color="Maroon" title="c4">pk</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">pk</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x2</font>
 <b>by </b><i><a class="txt" href="waybel_1.html#E2:89_1_2"><i><font color="Green" title="E61">A8</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>
<b>end;</b></div>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</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_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lk</font></span>)</span>;<br/><b>reconsider </b><font color="Maroon" title="c6">x'</font> = <font color="Maroon" title="c5">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="yellow_0.html#T59" title="YELLOW_0:th.59">YELLOW_0:59</a></i>;<br/><a NAME="E2:89_2"/><i><font color="Green" title="E61">A9</font></i>: 
<font color="Maroon" title="c4">pk</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x'</font>
 <b>by </b><i><a class="txt" href="waybel_1.html#E4:89"><i><font color="Green" title="E59">A4</font></i></a>, <a class="txt" href="waybel_1.html#E5:89"><i><font color="Green" title="E60">A5</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/><a NAME="E3:89_2"/>
<font color="Maroon" title="c5">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"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lk</font></span>)</span>
 ;<br/><a NAME="E4:89_2"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Lk</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="E5:89_2"/><b>then </b>
 ex <font color="Olive" title="b1">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c</font> &amp; <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> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">c</font> )
 <b>by </b><i><a class="txt" href="waybel_1.html#E3:89"><i><font color="Green" title="E58">A3</font></i></a></i>;<br/><a NAME="E6:89_2"/><b>then </b>
<font color="Maroon" title="c4">pk</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="txt" href="waybel_1.html#E2:89_2"><i><font color="Green" title="E61">A9</font></i></a>, <a class="ref" href="yellow_0.html#T61" title="YELLOW_0:th.61">YELLOW_0:61</a></i>;<br/><b>hence </b><a NAME="E7:89_2"/>
<font color="Maroon" title="c4">pk</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lk</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T91" title="TMAP_1:th.91">TMAP_1:91</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:89"/>
<font color="Maroon" title="c4">pk</font> <a href="yellow_2.html#R1" title="YELLOW_2:pred.1">&lt;=</a>  <a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lk</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_2.html#T10" title="YELLOW_2:th.10">YELLOW_2:10</a></i>;<br/>


</div>
