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

<a NAME="E1:39_1_1"/><i><font color="Green" title="E27">A2</font></i>: 
<font color="Maroon" title="c2">R</font> is   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="waybel20.html#E1:39"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="waybel20.html#D2" target="_self" title="WAYBEL20:def.2">Def2</a></i>;<br/>
<a NAME="E2:39_1_1"/><i><font color="Green" title="E28">A3</font></i>: 
 <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">R</font> is   <a href="waybel_5.html#NM3" title="WAYBEL_5:NM.3">CLSubFrame</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c1">L</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>
 
<b>by </b><i><a class="txt" href="waybel20.html#E1:39"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="waybel20.html#D2" target="_self" title="WAYBEL20:def.2">Def2</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> =  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p2">(<span class="default"><a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font></span>)</span>,$1</span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c3">k</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E4:39_1_1"/><i><font color="Green" title="E29">A4</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds  <font color="Maroon" title="c3">k</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></i>
<a NAME="E5:39_1_1"/><i><font color="Green" title="E30">A5</font></i>: 
<font color="Maroon" title="c2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font>
 
<b>by </b><i><a class="txt" href="waybel20.html#E1:39_1_1"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="waybel20.html#D1" target="_self" title="WAYBEL20:def.1">Def1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">k</font> = <font color="Maroon" title="c3">k</font> as   <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/>
<a NAME="E7:39_1_1"/><i><font color="Green" title="E31">A6</font></i>: 
<font color="Maroon" title="c4">k</font> is <a href="waybel_1.html#V6" title="WAYBEL_1:attr.6">projection</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_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 <font color="Maroon" title="c1">L</font>;<br/><b>set </b><font color="Maroon" title="c6">CRx</font> =  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font></span>)</span>,<font color="Maroon" title="c5">x</font>;<br/><a NAME="E1:39_1_1_1_1"/><i><font color="Green" title="E32">A7</font></i>: 
<font color="Maroon" title="c4">k</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>  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p2">(<span class="default"><a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font></span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel20.html#E4:39_1_1"><i><font color="Green" title="E29">A4</font></i></a></i>;<br/><a NAME="E2:39_1_1_1_1"/><b>then </b><i><font color="Green" title="E33">A8</font></i>: 
<font color="Maroon" title="c4">k</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p2">(<span class="default"><a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p3">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p4">(<span class="default"><a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font></span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="waybel20.html#E4:39_1_1"><i><font color="Green" title="E29">A4</font></i></a></i>;<br/><a NAME="E3:39_1_1_1_1"/>
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p3">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p4">(<span class="default"><a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font></span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">x</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 <b>by </b><i><a class="txt" href="waybel20.html#E1:39"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="waybel20.html#T36" target="_self" title="WAYBEL20:th.36">Th36</a></i>;<br/><a NAME="E4:39_1_1_1_1"/><b>then </b>
 <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p2">(<span class="default"><a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font></span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font></span>)</span>,<font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="txt" href="waybel20.html#E5:39_1_1"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="eqrel_1.html#T27" title="EQREL_1:th.27">EQREL_1:27</a></i>;<br/><a NAME="E5:39_1_1_1_1"/><b>then </b>
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p2">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p3">(<span class="default"><a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font></span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font></span>)</span>,<font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T31" title="EQREL_1:th.31">EQREL_1:31</a></i>;<br/><b>hence </b><a NAME="E6:39_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">k</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">k</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="waybel20.html#E1:39_1_1_1_1"><i><font color="Green" title="E32">A7</font></i></a>, <a class="txt" href="waybel20.html#E2:39_1_1_1_1"><i><font color="Green" title="E33">A8</font></i></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:39_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">k</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">k</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">k</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>hence </b><a NAME="E3:39_1_1_1"/>
<font color="Maroon" title="c4">k</font> is <a href="quantal1.html#V11" title="QUANTAL1:attr.11">idempotent</a>
 <b>by </b><i><a class="ref" href="quantal1.html#D9" title="QUANTAL1:def.9">QUANTAL1:def 9</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D13" title="WAYBEL_1:def.13">WAYBEL_1:def 13</a><br/>

<b>thus </b><a NAME="E4:39_1_1_1"/>
<font color="Maroon" title="c4">k</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="39_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <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>assume </b><a NAME="E1:39_1_1_1_2"/><i><font color="Green" title="E32">A9</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">y</font>
 ;<br/>

<i><font color="Green" title="E33">A10</font></i>:  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> = 
<font color="Maroon" title="c5">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c6">y</font>
<b>by </b><i><a class="ref" href="yellow_0.html#T40" title="YELLOW_0:th.40">YELLOW_0:40</a></i>
<br/>.= 
<font color="Maroon" title="c5">x</font>
<b>by </b><i><a class="txt" href="waybel20.html#E1:39_1_1_1_2"><i><font color="Green" title="E32">A9</font></i></a>, <a class="ref" href="yellow_0.html#T25" title="YELLOW_0:th.25">YELLOW_0:25</a></i>
;<br/>
<b>set </b><font color="Maroon" title="c7">CRx</font> =  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font></span>)</span>,<font color="Maroon" title="c5">x</font>;<br/>
<b>set </b><font color="Maroon" title="c8">CRy</font> =  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font></span>)</span>,<font color="Maroon" title="c6">y</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">SR</font> = <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p3">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p4">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p5">(<span class="default"><a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font></span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">x</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span>,<span class="p2"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p3">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p4">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p5">(<span class="default"><a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font></span>)</span>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">y</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c1">L</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span> ;<br/>
<a NAME="E4:39_1_1_1_2"/>
( <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p3">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p4">(<span class="default"><a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font></span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">x</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp; <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p3">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p4">(<span class="default"><a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font></span>)</span>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">y</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> )
 
<b>by </b><i><a class="txt" href="waybel20.html#E1:39"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="waybel20.html#T36" target="_self" title="WAYBEL20:th.36">Th36</a></i>;<br/>
<a NAME="E5:39_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c9">SR</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">R</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T38" title="ZFMISC_1:th.38">ZFMISC_1:38</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">SR'</font> = <font color="Maroon" title="c9">SR</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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="c2">R</font></span>)</span> <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="E7:39_1_1_1_2"/><i><font color="Green" title="E34">A11</font></i>: 
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c9">SR</font>,<span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c1">L</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17" title="YELLOW_0:th.17">YELLOW_0:17</a></i>;<br/>
<b>then </b><i><font color="Green" title="E35">A12</font></i>:  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c9">SR</font> = 
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p3">(<span class="default"><a href="yellow_3.html#K4" title="YELLOW_3:func.4">proj1</a> <font color="Maroon" title="c9">SR</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p3">(<span class="default"><a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Maroon" title="c9">SR</font></span>)</span></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span>
<b>by </b><i><a class="ref" href="waybel20.html#T7" target="_self" title="WAYBEL20:th.7">Th7</a></i>
<br/>.= 
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p4">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p5">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p0">(<span class="default"><a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font></span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p5">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p0">(<span class="default"><a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font></span>)</span>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p3">(<span class="default"><a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Maroon" title="c9">SR</font></span>)</span></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span>
<b>by </b><i><a class="ref" href="funct_5.html#T16" title="FUNCT_5:th.16">FUNCT_5:16</a></i>
<br/>.= 
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p4">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p5">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p0">(<span class="default"><a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font></span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p5">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p0">(<span class="default"><a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font></span>)</span>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span>
<b>by </b><i><a class="ref" href="funct_5.html#T16" title="FUNCT_5:th.16">FUNCT_5:16</a></i>
;<br/>
<a NAME="E9:39_1_1_1_2"/>
 <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon" title="c10">SR'</font>,<span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c1">L</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span> <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="c2">R</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel20.html#E2:39_1_1"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="waybel20.html#E7:39_1_1_1_2"><i><font color="Green" title="E34">A11</font></i></a>, <a class="ref" href="yellow_0.html#D18" title="YELLOW_0:def.18">YELLOW_0:def 18</a></i>;<br/>
<a NAME="E10:39_1_1_1_2"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p4">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p5">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p0">(<span class="default"><a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font></span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p5">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p0">(<span class="default"><a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font></span>)</span>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span>,<font color="Maroon" title="c5">x</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 
<b>by </b><i><a class="txt" href="waybel20.html#E2:39_1_1_1_2"><i><font color="Green" title="E33">A10</font></i></a>, <a class="txt" href="waybel20.html#E8:39_1_1_1_2"><i><font color="Green" title="E35">A12</font></i></a>, <a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></i>;<br/>
<a NAME="E11:39_1_1_1_2"/><b>then </b>
 <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p3">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p4">(<span class="default"><a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font></span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p3">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p4">(<span class="default"><a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font></span>)</span>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font></span>)</span>,<font color="Maroon" title="c5">x</font>
 
<b>by </b><i><a class="txt" href="waybel20.html#E5:39_1_1"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="eqrel_1.html#T27" title="EQREL_1:th.27">EQREL_1:27</a></i>;<br/>
<a NAME="E12:39_1_1_1_2"/><b>then </b><i><font color="Green" title="E36">A13</font></i>: 
 <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p2">(<span class="default"><a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font></span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p3">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p4">(<span class="default"><a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font></span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p3">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p4">(<span class="default"><a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font></span>)</span>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<b>by </b><i><a class="ref" href="yellow_2.html#T24" title="YELLOW_2:th.24">YELLOW_2:24</a></i>;<br/>
<a NAME="E13:39_1_1_1_2"/>
 <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p2">(<span class="default"><a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font></span>)</span>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p3">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p4">(<span class="default"><a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font></span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p3">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p4">(<span class="default"><a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font></span>)</span>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<a NAME="E14:39_1_1_1_2"/><b>then </b><i><font color="Green" title="E37">A14</font></i>: 
 <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p3">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p4">(<span class="default"><a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font></span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p3">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p4">(<span class="default"><a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font></span>)</span>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p2">(<span class="default"><a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font></span>)</span>,<font color="Maroon" title="c6">y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_2.html#T24" title="YELLOW_2:th.24">YELLOW_2:24</a></i>;<br/>
<a NAME="E15:39_1_1_1_2"/>
( <font color="Maroon" title="c4">k</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>  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p2">(<span class="default"><a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font></span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span> &amp; <font color="Maroon" title="c4">k</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p2">(<span class="default"><a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font></span>)</span>,<font color="Maroon" title="c6">y</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="waybel20.html#E4:39_1_1"><i><font color="Green" title="E29">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E16:39_1_1_1_2"/>
<font color="Maroon" title="c4">k</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="c4">k</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font>
 <b>by </b><i><a class="txt" href="waybel20.html#E12:39_1_1_1_2"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="waybel20.html#E14:39_1_1_1_2"><i><font color="Green" title="E37">A14</font></i></a>, <a class="ref" href="yellow_0.html#D2" title="YELLOW_0:def.2">YELLOW_0:def 2</a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_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 <font color="Maroon" title="c1">L</font>;<br/><a NAME="E1:39_1_1_2"/><i><font color="Green" title="E32">A15</font></i>: 
<font color="Maroon" title="c4">k</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>  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p2">(<span class="default"><a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font></span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel20.html#E4:39_1_1"><i><font color="Green" title="E29">A4</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c6">CRx</font> =  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font></span>)</span>,<font color="Maroon" title="c5">x</font>;<br/><a NAME="E2:39_1_1_2"/><i><font color="Green" title="E33">A16</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font></span>)</span>,<font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T28" title="EQREL_1:th.28">EQREL_1:28</a></i>;<br/><a NAME="E3:39_1_1_2"/>
 <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p2">(<span class="default"><a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font></span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font></span>)</span>,<font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T33" title="YELLOW_0:th.33">YELLOW_0:33</a></i>;<br/><a NAME="E4:39_1_1_2"/><b>then </b>
 <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p2">(<span class="default"><a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font></span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span> <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="waybel20.html#E2:39_1_1_2"><i><font color="Green" title="E33">A16</font></i></a>, <a class="ref" href="lattice3.html#D8" title="LATTICE3:def.8">LATTICE3:def 8</a></i>;<br/><b>hence </b><a NAME="E5:39_1_1_2"/>
<font color="Maroon" title="c4">k</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="struct_0.html#K3" title="STRUCT_0:func.3">id</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="c5">x</font>
 <b>by </b><i><a class="txt" href="waybel20.html#E1:39_1_1_2"><i><font color="Green" title="E32">A15</font></i></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:39_1_1"/><b>then </b>
<font color="Maroon" title="c4">k</font> <a href="yellow_2.html#R1" title="YELLOW_2:pred.1">&lt;=</a>  <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="yellow_2.html#T10" title="YELLOW_2:th.10">YELLOW_2:10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">k</font> = <font color="Maroon" title="c4">k</font> as  <a href="waybel_1.html#V8" title="WAYBEL_1:attr.8">kernel</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> <b>by </b><i><a class="txt" href="waybel20.html#E7:39_1_1"><i><font color="Green" title="E31">A6</font></i></a>, <a class="ref" href="waybel_1.html#D15" title="WAYBEL_1:def.15">WAYBEL_1:def 15</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">k</font>
;<br/>

<b>thus </b><a NAME="E11:39_1_1"/>
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds  <font color="Maroon" title="c5">k</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p2">(<span class="default"><a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font></span>)</span>,<font color="Olive" title="b1">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel20.html#E4:39_1_1"><i><font color="Green" title="E29">A4</font></i></a></i>;<br/>


</div>
