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

<b>let </b><font color="Maroon">M1</font> be   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="waybel_4.html#K8" target="_self">MonSet</a> <font color="Maroon">L</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_4.html#K3" target="_self">Aux</a> <font color="Maroon">L</font></span>)</span></span>)</span>, <font color="Maroon">M2</font> be   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="waybel_4.html#K8" target="_self">MonSet</a> <font color="Maroon">L</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_4.html#K3" target="_self">Aux</a> <font color="Maroon">L</font></span>)</span></span>)</span>;<br/>

<b>assume </b><a NAME="E1:68_1_2"/><i><font color="Green">E157</font></i>: 
for <font color="Olive">s</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">s</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_4.html#K8" target="_self">MonSet</a> <font color="Maroon">L</font></span>)</span> holds <br/> ex <font color="Olive">AR</font> being <a href="waybel_4.html#V5" target="_self">auxiliary</a> <a href="alg_1.html#NM1">Relation</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">AR</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font> &amp; ( for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">AR</font> iff  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> ex <font color="Olive">s'</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span> st <br/>( <font color="Olive">x'</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> &amp; <font color="Olive">y'</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> &amp; <font color="Olive">s'</font> <a href="hidden.html#R1">=</a> <font color="Olive">s</font> &amp; <font color="Olive">x'</font> <a href="hidden.html#R2">in</a> <font color="Olive">s'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y'</font> ) ) ) )
 ;<br/>

<b>assume </b><a NAME="E2:68_1_2"/><i><font color="Green">E158</font></i>: 
for <font color="Olive">s</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">s</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_4.html#K8" target="_self">MonSet</a> <font color="Maroon">L</font></span>)</span> holds <br/> ex <font color="Olive">AR</font> being <a href="waybel_4.html#V5" target="_self">auxiliary</a> <a href="alg_1.html#NM1">Relation</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">AR</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font> &amp; ( for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">AR</font> iff  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> ex <font color="Olive">s'</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span> st <br/>( <font color="Olive">x'</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> &amp; <font color="Olive">y'</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> &amp; <font color="Olive">s'</font> <a href="hidden.html#R1">=</a> <font color="Olive">s</font> &amp; <font color="Olive">x'</font> <a href="hidden.html#R2">in</a> <font color="Olive">s'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y'</font> ) ) ) )
 ;<br/>

<a NAME="E3:68_1_2"/>
<font color="Maroon">M1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">M2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:68_1_2_1"/><i><font color="Green">E159</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">M1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_4.html#K8" target="_self">MonSet</a> <font color="Maroon">L</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:68_1_2_1"/><i><font color="Green">E160</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">M2</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_4.html#K8" target="_self">MonSet</a> <font color="Maroon">L</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:68_1_2_1"/>
for <font color="Olive">s</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">s</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_4.html#K8" target="_self">MonSet</a> <font color="Maroon">L</font></span>)</span> holds <br/><font color="Maroon">M1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:68_1_2_1_1"/><i><font color="Green">E161</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_4.html#K8" target="_self">MonSet</a> <font color="Maroon">L</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">AR1</font> being  <a href="waybel_4.html#V5" target="_self">auxiliary</a> <a href="alg_1.html#NM1">Relation</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:68_1_2_1_1"/><i><font color="Green">E163</font></i>: 
( <font color="Maroon">AR1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> &amp; ( for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">AR1</font> iff  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> ex <font color="Olive">s'</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span> st <br/>( <font color="Olive">x'</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> &amp; <font color="Olive">y'</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> &amp; <font color="Olive">s'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; <font color="Olive">x'</font> <a href="hidden.html#R2">in</a> <font color="Olive">s'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y'</font> ) ) ) )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">AR2</font> being  <a href="waybel_4.html#V5" target="_self">auxiliary</a> <a href="alg_1.html#NM1">Relation</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E5:68_1_2_1_1"/><i><font color="Green">E165</font></i>: 
( <font color="Maroon">AR2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> &amp; ( for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">AR2</font> iff  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> ex <font color="Olive">s'</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span> st <br/>( <font color="Olive">x'</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> &amp; <font color="Olive">y'</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> &amp; <font color="Olive">s'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; <font color="Olive">x'</font> <a href="hidden.html#R2">in</a> <font color="Olive">s'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y'</font> ) ) ) )
 <b>by </b><i>, <a class="ref" href="waybel_4.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E6:68_1_2_1_1"/>
<font color="Maroon">AR1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">AR2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_2_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_1.html#D2">RELAT_1:def 2</a><br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:68_1_2_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">AR1</font>
 ;<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>, <font color="Maroon">s'</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:68_1_2_1_1_1_1"/><i><font color="Green">E166</font></i>: 
( <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">s'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">s'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y'</font> )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E4:68_1_2_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">AR2</font>
 <b>by </b><i>, <a class="ref" href="waybel_4.html#D6" target="_self">Def6</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:68_1_2_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">AR2</font>
 ;<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>, <font color="Maroon">s'</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E4:68_1_2_1_1_1"/><i><font color="Green">E167</font></i>: 
( <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">s'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">s'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y'</font> )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E5:68_1_2_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">AR1</font>
 <b>by </b><i>, <a class="ref" href="waybel_4.html#D6" target="_self">Def6</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:68_1_2_1_1"/>
<font color="Maroon">M1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:68_1_2_1"/>
<font color="Maroon">M1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">M2</font>
 <b>by </b><i><a class="txt" href="waybel_4.html#E4"><i><font color="Green">Lemma41</font></i></a>, <a class="ref" href="waybel_4.html#D4" target="_self">Def4</a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:68_1_2"/>
<font color="Maroon">M1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M2</font>
 ;<br/>


</div>
