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

<b>let </b><font color="Maroon">L</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_3.html#V3">continuous</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">R</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">L</font>,<font color="Maroon">L</font></span><a href="yellow_3.html#K3">:]</a></span>;<br/><b>let </b><font color="Maroon">k</font> be  <a href="waybel_1.html#V8">kernel</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L</font>,<font color="Maroon">L</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:41"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">k</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 <b>and </b><br/><a NAME="E2:41"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><font color="Maroon">k</font>,<font color="Maroon">k</font></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span>)</span>
 ;<br/>

<a NAME="E3:41"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">R</font> is   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
<b>by </b><i><a class="txt" href="waybel20.html#E1:41"><i><font color="Green">E30</font></i></a>, </i>;<br/>
<b>set </b><font color="Maroon">ER</font> =  <a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font>;<br/>
<a NAME="E4:41"/><i><font color="Green">E39</font></i>: 
 <a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="txt" href="waybel20.html#E2:41"><i><font color="Green">E31</font></i></a>, </i>;<br/>
<b>set </b><font color="Maroon">cL</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>;<br/>
<a NAME="E5:41"/><i><font color="Green">E40</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:41"/><b>then </b><i><font color="Green">E41</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><font color="Maroon">k</font>,<font color="Maroon">k</font></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_3.html#D9">FUNCT_3:def 9</a></i>;<br/>
<b>consider </b><font color="Maroon">xx</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>
<a NAME="E8:41"/>
 <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">xx</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">CER</font> =  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span> as  non <a href="xboole_0.html#V1">empty</a> <a href="setfam_1.html#NM1">Subset-Family</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Olive">x</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Olive">y</font> &amp; <font color="Maroon">k</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">k</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> );<br/>
<b>consider </b><font color="Maroon">LR</font> being  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a> <b> such that </b><br/><a NAME="E11:41"/><i><font color="Green">E42</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">LR</font> <a href="hidden.html#R1">=</a> <font color="Maroon">CER</font>
 <b>and </b><br/><a NAME="E12:41"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">LR</font> holds <br/> ( <font color="Olive">a</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>,<font color="Olive">b</font>] )
 <b>from </b><i><a class="ref" href="yellow_0.html#S1">YELLOW_0:sch 1</a>();<br/></i>
<a NAME="E13:41"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> holds <br/> (  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Olive">b</font> iff <font color="Maroon">k</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>, <font color="Maroon">b</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:41_1_1"/>
 <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">b</font>
 ;<br/><a NAME="E2:41_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T31">EQREL_1:31</a></i>;<br/><a NAME="E3:41_1_1"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="waybel20.html#T2" target="_self">Th2</a>, <a class="ref" href="eqrel_1.html#T27">EQREL_1:27</a></i>;<br/><a NAME="E4:41_1_1"/><b>then </b>
<span class="p1"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><font color="Maroon">k</font>,<font color="Maroon">k</font></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 <b>by </b><i><a class="txt" href="waybel20.html#E1:41"><i><font color="Green">E30</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/><a NAME="E5:41_1_1"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="funct_3.html#D9">FUNCT_3:def 9</a></i>;<br/><b>hence </b><a NAME="E6:41_1_1"/>
<font color="Maroon">k</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D10">RELAT_1:def 10</a></i>;<br/></div>
<b>end;</b></div>

<a NAME="E2:41_1"/><i><font color="Green">E46</font></i>: 
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>assume </b><a NAME="E3:41_1"/>
<font color="Maroon">k</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>
 ;<br/>

<a NAME="E4:41_1"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#D10">RELAT_1:def 10</a></i>;<br/>
<a NAME="E5:41_1"/><b>then </b>
<span class="p1"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><font color="Maroon">k</font>,<font color="Maroon">k</font></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
<b>by </b><i>, <a class="ref" href="funct_3.html#D9">FUNCT_3:def 9</a></i>;<br/>
<a NAME="E6:41_1"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a>  <a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="txt" href="waybel20.html#E1:41"><i><font color="Green">E30</font></i></a>, <a class="ref" href="waybel20.html#T2" target="_self">Th2</a>, , , <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E7:41_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T27">EQREL_1:27</a></i>;<br/>
<b>hence </b><a NAME="E8:41_1"/>
 <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T31">EQREL_1:31</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">rngk</font> =  <a href="relset_1.html#K5">rng</a> <font color="Maroon">k</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Olive">a</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font> );<br/>
<a NAME="E15:41"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">CER</font>  ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">rngk</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">CER</font>;<br/>

<b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:41_2"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i><a class="txt" href="waybel20.html#E4:41"><i><font color="Green">E39</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">ky</font> = <font color="Maroon">k</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">rngk</font> <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>take </b>
<font color="Maroon">ky</font>
;<br/>

<b>thus </b><a NAME="E5:41_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">ky</font>]
 <b>by </b><i><a class="txt" href="waybel20.html#E4:41"><i><font color="Green">E39</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">CER</font>,<font color="Maroon">rngk</font><b> such that </b><br/><a NAME="E17:41"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">CER</font> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i>
<a NAME="E18:41"/><i><font color="Green">E52</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">CER</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E19:41"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p2">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Olive">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<b>reconsider </b><font color="Maroon">CERx</font> =  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">CER</font> <b>by </b><i><a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">a</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:41_3"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">CERx</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">a</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">CERx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="txt" href="waybel20.html#E4:41"><i><font color="Green">E39</font></i></a></i>;<br/>
<b>thus </b><a NAME="E4:41_3"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p2">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="waybel20.html#E3:41"><i><font color="Green">E37</font></i></a>, <a class="txt" href="waybel20.html#E11:41"><i><font color="Green">E42</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">tIR</font> = the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">LR</font>;<br/>
<b>set </b><font color="Maroon">IR</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p4">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Olive">x</font></span>)</span>,<span class="p3">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p4">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Olive">y</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Maroon">k</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">k</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> </span>}</span> ;<br/>
<a NAME="E20:41"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">LR</font>  ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Olive">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">LR</font>;<br/>

<a NAME="E1:41_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CER</font>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:41_4"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">a</font>
;<br/>

<b>thus </b><a NAME="E5:41_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">a</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:41_5"/><i><font color="Green">E119</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CER</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CER</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> )
 ;<br/><b>then reconsider </b><font color="Maroon">x1'</font> = <font color="Maroon">x1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">LR</font> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">x2'</font> = <font color="Maroon">x2</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">LR</font> <b>by </b><i>, </i>;<br/><b>consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E5:41_5"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">x1'</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">a</font>
 <b>by </b><i><a class="txt" href="waybel20.html#E11:41"><i><font color="Green">E42</font></i></a></i>;<br/><b>consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E7:41_5"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">x2'</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">b</font>
 <b>by </b><i><a class="txt" href="waybel20.html#E11:41"><i><font color="Green">E42</font></i></a></i>;<br/><a NAME="E8:41_5"/>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="txt" href="waybel20.html#E6:41"><i><font color="Green">E41</font></i></a>, <a class="txt" href="waybel20.html#E12:41"><i><font color="Green">E44</font></i></a>, <a class="txt" href="waybel20.html#E13:41"><i><font color="Green">E45</font></i></a></i>;<br/><b>hence </b><a NAME="E9:41_5"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i><a class="txt" href="waybel20.html#E3:41"><i><font color="Green">E37</font></i></a>, , <a class="txt" href="waybel20.html#E12:41"><i><font color="Green">E44</font></i></a>, <a class="txt" href="waybel20.html#E13:41"><i><font color="Green">E45</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E22:41"/><b>then </b><i><font color="Green">E122</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="funct_2.html#T25">FUNCT_2:25</a></i>;<br/>
<a NAME="E23:41"/><i><font color="Green">E123</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">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">rngk</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">LR</font>,<span class="p1">(<span class="default"><a href="yellow_2.html#K1">Image</a> <font color="Maroon">k</font></span>)</span> <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:41_6_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:41_6_1"/><i><font color="Green">E124</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">LR</font> <b>by </b><i><a class="txt" href="waybel20.html#E13:41"><i><font color="Green">E45</font></i></a></i>;<br/><b>consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E6:41_6_1"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">a</font>
 <b>by </b><i><a class="txt" href="waybel20.html#E11:41"><i><font color="Green">E42</font></i></a></i>;<br/><a NAME="E7:41_6_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="txt" href="waybel20.html#E6:41"><i><font color="Green">E41</font></i></a>, </i>;<br/><b>hence </b><a NAME="E8:41_6_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">rngk</font>
 <b>by </b><i>, <a class="txt" href="waybel20.html#E13:41"><i><font color="Green">E45</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:41_6"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">rngk</font>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:41_6"/><i><font color="Green">E145</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">k</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i><a class="txt" href="waybel20.html#E13:41"><i><font color="Green">E45</font></i></a></i>;<br/><a NAME="E6:41_6"/><i><font color="Green">E146</font></i>: 
 <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CER</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/><a NAME="E7:41_6"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p2">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="waybel20.html#E6:41"><i><font color="Green">E41</font></i></a></i>;<br/><b>hence </b><a NAME="E8:41_6"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="waybel20.html#E5:41"><i><font color="Green">E40</font></i></a>, <a class="txt" href="waybel20.html#E13:41"><i><font color="Green">E45</font></i></a>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div>
<a NAME="E26:41"/><b>then </b><i><font color="Green">E147</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">rngk</font>
 
<b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<a NAME="E27:41"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">LR</font> holds <br/> ( <font color="Olive">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">y</font> iff <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">LR</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">LR</font>;<br/>

<a NAME="E1:41_7"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CER</font>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:41_7"/><i><font color="Green">E150</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i/>;<br/>
<a NAME="E5:41_7"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CER</font>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:41_7"/><i><font color="Green">E151</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i><a class="txt" href="waybel20.html#E15:41"><i><font color="Green">E47</font></i></a></i>;<br/>
<a NAME="E9:41_7"/><i><font color="Green">E152</font></i>: 
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> )
 
<b>by </b><i><a class="txt" href="waybel20.html#E6:41"><i><font color="Green">E41</font></i></a>, , <a class="txt" href="waybel20.html#E15:41"><i><font color="Green">E47</font></i></a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:41_7_1"/>
<font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font>
 ;<br/><b>then consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">d</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:41_7_1"/><i><font color="Green">E153</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">c</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">d</font> &amp; <font color="Maroon">k</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">k</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">d</font> )
 <b>by </b><i/>;<br/><a NAME="E4:41_7_1"/>
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">d</font> )
 <b>by </b><i><a class="txt" href="waybel20.html#E6:41"><i><font color="Green">E41</font></i></a>, </i>;<br/><b>hence </b><a NAME="E5:41_7_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T61">YELLOW_0:61</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E11:41_7"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E12:41_7"/><b>then </b>
<font color="Maroon">k</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">k</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#T60">YELLOW_0:60</a></i>;<br/>
<b>hence </b><a NAME="E13:41_7"/>
<font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font>
 <b>by </b><i>, , <a class="txt" href="waybel20.html#E15:41"><i><font color="Green">E47</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E28:41"/><b>then </b><i><font color="Green">E154</font></i>: 
<font color="Maroon">f</font> is <a href="waybel_0.html#V23">isomorphic</a>
 
<b>by </b><i>, <a class="txt" href="waybel20.html#E12:41"><i><font color="Green">E44</font></i></a>, <a class="txt" href="waybel20.html#E13:41"><i><font color="Green">E45</font></i></a>, <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/>
<a NAME="E29:41"/><b>then </b>
<font color="Maroon">LR</font>, <a href="yellow_2.html#K1">Image</a> <font color="Maroon">k</font> <a href="waybel_1.html#R5">are_isomorphic</a> 
 
<b>by </b><i><a class="ref" href="waybel_1.html#D8">WAYBEL_1:def 8</a></i>;<br/>
<a NAME="E30:41"/><b>then </b><i><font color="Green">E155</font></i>: 
 <a href="yellow_2.html#K1">Image</a> <font color="Maroon">k</font>,<font color="Maroon">LR</font> <a href="waybel_1.html#R5">are_isomorphic</a> 
 
<b>by </b><i><a class="ref" href="waybel_1.html#T7">WAYBEL_1:7</a></i>;<br/>
<a NAME="E31:41"/><i><font color="Green">E156</font></i>: 
 <a href="yellow_2.html#K1">Image</a> <font color="Maroon">k</font> is  <a href="waybel_3.html#V3">continuous</a> <a href="waybel_0.html#NM8">LATTICE</a>
 
<b>by </b><i>, <a class="ref" href="waybel15.html#T16">WAYBEL15:16</a></i>;<br/>
<b>reconsider </b><font color="Maroon">LR</font> = <font color="Maroon">LR</font> as  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V1">strict</a> <a href="orders_2.html#NM1">Poset</a> <b>by </b><i><a class="txt" href="waybel20.html#E15:41"><i><font color="Green">E47</font></i></a>, , , </i>;<br/>
<a NAME="E33:41"/>
 <a href="yellow_2.html#K1">Image</a> <font color="Maroon">k</font> is <a href="lattice3.html#V3">complete</a>
 
<b>by </b><i><a class="ref" href="waybel_1.html#T57">WAYBEL_1:57</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">LR</font> = <font color="Maroon">LR</font> as  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V1">strict</a> <a href="lattice3.html#V3">complete</a> <a href="orders_2.html#NM1">Poset</a> <b>by </b><i><a class="txt" href="waybel20.html#E15:41"><i><font color="Green">E47</font></i></a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">LR</font> = <font color="Maroon">LR</font> as  <a href="orders_2.html#V1">strict</a> <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a> ;<br/>
<b>reconsider </b><font color="Maroon">LR</font> = <font color="Maroon">LR</font> as  <a href="orders_2.html#V1">strict</a> <a href="lattice3.html#V3">complete</a> <a href="waybel_3.html#V3">continuous</a> <a href="waybel_0.html#NM8">LATTICE</a> <b>by </b><i><a class="txt" href="waybel20.html#E15:41"><i><font color="Green">E47</font></i></a>, , <a class="ref" href="waybel15.html#T11">WAYBEL15:11</a></i>;<br/>
<b>take </b>
<font color="Maroon">LR</font>
;<br/>

<b>thus </b><a NAME="E37:41"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">LR</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>
 <b>by </b><i/>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:41_8_1"/><i><font color="Green">E157</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">LR</font>
 ;<br/><b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:41_8_1"/><i><font color="Green">E170</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CER</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CER</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font>, <font color="Maroon">b</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">LR</font> <b>by </b><i>, <a class="txt" href="waybel20.html#E17:41"><i><font color="Green">E50</font></i></a></i>;<br/><a NAME="E5:41_8_1"/>
<font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="txt" href="waybel20.html#E17:41"><i><font color="Green">E50</font></i></a>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/><b>then consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E7:41_8_1"/><i><font color="Green">E171</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">x</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">y</font> &amp; <font color="Maroon">k</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">k</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E8:41_8_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p4">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Olive">x</font></span>)</span>,<span class="p3">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p4">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Olive">y</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Maroon">k</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">k</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="waybel20.html#E17:41"><i><font color="Green">E50</font></i></a>, </i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:41_8"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p4">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Olive">x</font></span>)</span>,<span class="p3">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p4">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Olive">y</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Maroon">k</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">k</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E4:41_8"/><i><font color="Green">E172</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p3">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p3">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">y</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">k</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">k</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> )
 ;<br/><b>reconsider </b><font color="Maroon">a</font> =  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">LR</font> <b>by </b><i>, <a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">b</font> =  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">LR</font> <b>by </b><i>, <a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/><a NAME="E7:41_8"/>
<font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E8:41_8"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">LR</font>
 <b>by </b><i>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E39:41"/>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">LR</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p4">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Olive">x</font></span>)</span>,<span class="p3">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p4">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Olive">y</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Maroon">k</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">k</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>

<b>let </b><font color="Maroon">g</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L</font>,<font color="Maroon">LR</font>;<br/>

<b>assume </b><a NAME="E40:41"/><i><font color="Green">E173</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Olive">x</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">f</font> <a href="funct_1.html#K2">"</a>  as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1">Image</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">LR</font> <b>by </b><i>, <a class="txt" href="waybel20.html#E12:41"><i><font color="Green">E44</font></i></a>, <a class="txt" href="waybel20.html#E13:41"><i><font color="Green">E45</font></i></a>, <a class="ref" href="funct_2.html#T31">FUNCT_2:31</a></i>;<br/>
<a NAME="E42:41"/><i><font color="Green">E175</font></i>: 
<font color="Maroon">f'</font> is <a href="waybel_0.html#V23">isomorphic</a>
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#T68">WAYBEL_0:68</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:41_9"/><i><font color="Green">E176</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 ;<br/><b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/><a NAME="E3:41_9"/><i><font color="Green">E177</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K2">corestr</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E4:41_9"/><i><font color="Green">E178</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p2">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">x'</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font>
 <b>by </b><i><a class="txt" href="waybel20.html#E6:41"><i><font color="Green">E41</font></i></a></i>;<br/><a NAME="E5:41_9"/><i><font color="Green">E179</font></i>: 
 <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CER</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="waybel_1.html#K2">corestr</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">f'</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_1.html#K2">corestr</a> <font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="txt" href="waybel20.html#E18:41"><i><font color="Green">E52</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">f'</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="waybel_1.html#T32">WAYBEL_1:32</a></i>
<br/>.= 
 <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">x'</font>
<b>by </b><i><a class="txt" href="waybel20.html#E5:41"><i><font color="Green">E40</font></i></a>, , <a class="txt" href="waybel20.html#E19:41"><i><font color="Green">E53</font></i></a>, , <a class="ref" href="funct_1.html#T54">FUNCT_1:54</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<a NAME="E44:41"/><b>then </b><i><font color="Green">E180</font></i>: 
<font color="Maroon">g</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">f'</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K2">corestr</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T18">FUNCT_2:18</a></i>;<br/>
<a NAME="E45:41"/><i><font color="Green">E181</font></i>: 
 <a href="waybel_1.html#K2">corestr</a> <font color="Maroon">k</font> is <a href="waybel_0.html#V17">infs-preserving</a>
 
<b>by </b><i><a class="ref" href="waybel_1.html#T59">WAYBEL_1:59</a></i>;<br/>
<a NAME="E46:41"/><i><font color="Green">E182</font></i>: 
 <a href="waybel_1.html#K2">corestr</a> <font color="Maroon">k</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E47:41"/><i><font color="Green">E183</font></i>: 
<font color="Maroon">f'</font> is <a href="waybel_0.html#V17">infs-preserving</a>
 
<b>by </b><i><a class="txt" href="waybel20.html#E17:41"><i><font color="Green">E50</font></i></a>, <a class="ref" href="waybel13.html#T20">WAYBEL13:20</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">f'</font> as  <a href="waybel_0.html#V18">sups-preserving</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1">Image</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">LR</font> <b>by </b><i><a class="txt" href="waybel20.html#E17:41"><i><font color="Green">E50</font></i></a>, <a class="ref" href="waybel13.html#T20">WAYBEL13:20</a></i>;<br/>
<a NAME="E49:41"/><i><font color="Green">E184</font></i>: 
<font color="Maroon">f'</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 
;<br/>
<a NAME="E50:41"/><i><font color="Green">E185</font></i>: 
<font color="Maroon">g</font> is <a href="waybel_0.html#V17">infs-preserving</a>
 
<b>by </b><i>, <a class="txt" href="waybel20.html#E18:41"><i><font color="Green">E52</font></i></a>, , </i>;<br/>
<a NAME="E51:41"/>
<font color="Maroon">g</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 
<b>by </b><i>, <a class="txt" href="waybel20.html#E19:41"><i><font color="Green">E53</font></i></a>, <a class="txt" href="waybel20.html#E20:41"><i><font color="Green">E55</font></i></a>, <a class="ref" href="waybel15.html#T13">WAYBEL15:13</a></i>;<br/>
<b>hence </b><a NAME="E52:41"/>
<font color="Maroon">g</font> is    <a href="waybel16.html#M1">CLHomomorphism</a> of <font color="Maroon">L</font>,<font color="Maroon">LR</font>
 <b>by </b><i>, <a class="ref" href="waybel16.html#D1">WAYBEL16:def 1</a></i>;<br/>


</div>
