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<b>let </b><font color="Maroon">L</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">mp</font> be   <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>;<br/>



<b>assume </b><a NAME="E1:100"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">mp</font> is <a href="waybel_4.html#V7" target="_self">approximating</a> &amp; <font color="Maroon">mp</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">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><b> such that </b><br/><a NAME="E3:100"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">AR</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K10" target="_self">Map2Rel</a> <font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">mp</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">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="funct_2.html#R2">=</a> <font color="Maroon">mp</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1"><b>now </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>consider </b><font color="Maroon">ii</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E2:100_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">ii</font> <a href="hidden.html#R1">=</a> <font color="Maroon">mp</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">ii</font> )
 <b>by </b><i><a class="ref" href="waybel_4.html#T2" target="_self">Th2</a>, </i>;<br/><a NAME="E3:100_1"/>
<font color="Maroon">AR</font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">mp</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:100_1_1"/>
<font color="Maroon">AR</font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">mp</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:100_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">AR</font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">x</font>
 ;<br/>

<a NAME="E2:100_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">AR</font>
 
<b>by </b><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>, <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="E4:100_1_1_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">s'</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">mp</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="E5:100_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">mp</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="waybel_4.html#T4" target="_self">Th4</a></i>;<br/>


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

<b>thus </b><a NAME="E2:100_1_1"/>
<font color="Maroon">mp</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">AR</font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:100_1_1_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">mp</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 ;<br/>

<b>then 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><a class="ref" href="waybel_4.html#D10" target="_self">Def10</a></i>;<br/>
<a NAME="E3:100_1_1_2"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a'</font>,<font color="Maroon">x</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">AR</font>
 
<b>by </b><i>, <a class="ref" href="waybel_4.html#T4" target="_self">Th4</a></i>;<br/>
<b>hence </b><a NAME="E4:100_1_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">AR</font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">x</font>
 ;<br/>


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


</div><b>end;</b></div><b>hence </b><a NAME="E4:100_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">AR</font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_4.html#D10" target="_self">Def10</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">AR</font> = <font color="Maroon">AR</font> as  <a href="waybel_4.html#V5" target="_self">auxiliary</a> <a href="waybel_4.html#V6" target="_self">approximating</a> <a href="alg_1.html#NM1">Relation</a> of <font color="Maroon">L</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">AR</font>
;<br/>

<b>thus </b><a NAME="E6:100"/>
<font color="Maroon">AR</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K10" target="_self">Map2Rel</a> <font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">mp</font>
 <b>by </b><i/>;<br/>


</div>
