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<b>let </b><font color="Maroon">R1</font> be   <a href="relat_1.html#NM1">Relation</a>, <font color="Maroon">R2</font> be   <a href="relat_1.html#NM1">Relation</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:83_1_2"/><i><font color="Green">E42</font></i>: 
 <a href="relat_1.html#K3">field</a> <font color="Maroon">R1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E2:83_1_2"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds <br/> ( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">p</font>,<font color="Olive">q</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> iff <font color="Olive">p</font> <a href="filter_0.html#K9" target="_self">&lt;=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 <b>and </b><br/><a NAME="E3:83_1_2"/><i><font color="Green">E44</font></i>: 
 <a href="relat_1.html#K3">field</a> <font color="Maroon">R2</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E4:83_1_2"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds <br/> ( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">p</font>,<font color="Olive">q</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R2</font> iff <font color="Olive">p</font> <a href="filter_0.html#K9" target="_self">&lt;=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 ;<br/>

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



<b>thus </b><a NAME="E5:83_1_2"/>
( <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">R1</font> implies <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">R2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:83_1_2_1"/><i><font color="Green">E60</font></i>: 
<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">R1</font>
 ;<br/>

<a NAME="E2:83_1_2_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R1</font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">x</font>, <font color="Maroon">q</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="ref" href="filter_0.html#T22" target="_self">Th22</a></i>;<br/>
<a NAME="E4:83_1_2_1"/>
<font color="Maroon">p</font> <a href="filter_0.html#K9" target="_self">&lt;=&gt;</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="txt" href="filter_0.html#E6:83_1_2"><i><font color="Green">E61</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:83_1_2_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">R2</font>
 <b>by </b><i><a class="txt" href="filter_0.html#E1:83_1_2_1"><i><font color="Green">E60</font></i></a></i>;<br/>


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

<b>assume </b><a NAME="E6:83_1_2"/><i><font color="Green">E61</font></i>: 
<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">R2</font>
 ;<br/>

<a NAME="E7:83_1_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R2</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R2</font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">x</font>, <font color="Maroon">q</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="filter_0.html#E4:83_1_2"><i><font color="Green">E59</font></i></a></i>;<br/>
<a NAME="E9:83_1_2"/>
<font color="Maroon">p</font> <a href="filter_0.html#K9" target="_self">&lt;=&gt;</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="txt" href="filter_0.html#E6:83_1_2"><i><font color="Green">E61</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:83_1_2"/>
<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">R1</font>
 <b>by </b><i/>;<br/>


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