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

<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>;<br/><b>let </b><font color="Maroon" title="c2">R</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 <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/>



<b>assume </b><a NAME="E1:38"/><i><font color="Green" title="E26">A1</font></i>: 
<font color="Maroon" title="c2">R</font> is <a href="waybel20.html#V1" title="WAYBEL20:attr.1">CLCongruence</a>
 ;<br/>

<a NAME="E2:38"/><b>then </b><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="ref" href="waybel20.html#D2" target="_self" title="WAYBEL20:def.2">Def2</a></i>;<br/>
<a NAME="E3:38"/><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#E1:38"><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="E4:38"/><i><font color="Green" title="E29">A4</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#E2:38"><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>let </b><font color="Maroon" title="c3">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="c4">CRx</font> =  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">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="c3">x</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">SR</font> = <span class="p1"><a href="yellow_3.html#K6" title="YELLOW_3:func.6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="eqrel_1.html#K7" title="EQREL_1:func.7">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="c3">x</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="yellow_3.html#K6" title="YELLOW_3:func.6">:]</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="E6:38"/>
<font color="Maroon" title="c5">SR</font> <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="c2">R</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:38_1"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">SR</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">a1</font>, <font color="Maroon" title="c8">a2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:38_1"/><i><font color="Green" title="E30">A5</font></i>: 
( <font color="Maroon" title="c7">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">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="c3">x</font> &amp; <font color="Maroon" title="c8">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E4:38_1"/>
<font color="Maroon" title="c8">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font>
 
<b>by </b><i><a class="txt" href="waybel20.html#E3:38_1"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:38_1"/><b>then </b>
<font color="Maroon" title="c6">a</font> <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#E4:38"><i><font color="Green" title="E29">A4</font></i></a>, <a class="txt" href="waybel20.html#E3:38_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/>
<b>hence </b><a NAME="E6:38_1"/>
<font color="Maroon" title="c6">a</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="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/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c6">SR'</font> = <font color="Maroon" title="c5">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> ;<br/>
<a NAME="E8:38"/><i><font color="Green" title="E30">A6</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">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="c3">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="E9:38"/><i><font color="Green" title="E31">A7</font></i>: 
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c5">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="E32">A8</font></i>:  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c5">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="c5">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="c5">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">(<span class="default"><a href="eqrel_1.html#K7" title="EQREL_1:func.7">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="c3">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="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Maroon" title="c5">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#T11" title="FUNCT_5:th.11">FUNCT_5:11</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">(<span class="default"><a href="eqrel_1.html#K7" title="EQREL_1:func.7">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="c3">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"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span>
<b>by </b><i><a class="txt" href="waybel20.html#E8:38"><i><font color="Green" title="E30">A6</font></i></a>, <a class="ref" href="funct_5.html#T11" title="FUNCT_5:th.11">FUNCT_5:11</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">(<span class="default"><a href="eqrel_1.html#K7" title="EQREL_1:func.7">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="c3">x</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">x</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span>
<b>by </b><i><a class="ref" href="yellow_0.html#T39" title="YELLOW_0:th.39">YELLOW_0:39</a></i>
;<br/>
<a NAME="E11:38"/>
 <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon" title="c6">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#E3:38"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="waybel20.html#E9:38"><i><font color="Green" title="E31">A7</font></i></a>, <a class="ref" href="yellow_0.html#D18" title="YELLOW_0:def.18">YELLOW_0:def 18</a></i>;<br/>
<b>hence </b><a NAME="E12:38"/>
<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#K7" title="EQREL_1:func.7">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="c3">x</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">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#E10:38"><i><font color="Green" title="E32">A8</font></i></a>, <a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></i>;<br/>


</div>
