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

<b>consider </b><font color="Maroon">Y</font> being    <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E2:20_1"/><i><font color="Green">E49</font></i>: 
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">Y</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] ) )
 <b>from </b><i><a class="ref" href="relset_1.html#S1">RELSET_1:sch 1</a>();<br/></i>
<a NAME="E3:20_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">Y</font> <a href="relat_2.html#R1">is_reflexive_in</a> <font color="Maroon">F<sub>1</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D1">RELAT_2:def 1</a><br/>

<b>assume </b><a NAME="E1:20_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/>

<a NAME="E2:20_1_1"/><b>then </b>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">x</font>]
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:20_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:20_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">Y</font> <a href="relat_2.html#R3">is_symmetric_in</a> <font color="Maroon">F<sub>1</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_2"><b>proof </b></a><div class="add">

<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_2.html#D3">RELAT_2:def 3</a><br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:20_1_2"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <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">Y</font> )
 ;<br/>

<a NAME="E2:20_1_2"/><b>then </b>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>]
 
<b>by </b><i/>;<br/>
<a NAME="E3:20_1_2"/><b>then </b>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">y</font>,<font color="Maroon">x</font>] )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:20_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E5:20_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">Y</font> <a href="relat_2.html#R8">is_transitive_in</a> <font color="Maroon">F<sub>1</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_3"><b>proof </b></a><div class="add">

<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_2.html#D8">RELAT_2:def 8</a><br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/>





<b>assume </b><a NAME="E1:20_1_3"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <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">Y</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 ;<br/>

<a NAME="E2:20_1_3"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>]
 
<b>by </b><i/>;<br/>
<a NAME="E3:20_1_3"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">y</font>,<font color="Maroon">z</font>]
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:20_1_3"/><b>then </b>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">z</font>]
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:20_1_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:20_1"/><i><font color="Green">E59</font></i>: 
 <a href="relat_1.html#K3">field</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i>, <a class="ref" href="orders_1.html#T98">ORDERS_1:98</a></i>;<br/>
<a NAME="E7:20_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i>, <a class="ref" href="orders_1.html#T98">ORDERS_1:98</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">R1</font> = <font color="Maroon">Y</font> as   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i>, , , <a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a>, <a class="ref" href="relat_2.html#D11">RELAT_2:def 11</a>, <a class="ref" href="relat_2.html#D16">RELAT_2:def 16</a></i>;<br/>
<b>take </b>
<font color="Maroon">R1</font>
;<br/>

<b>thus </b><a NAME="E9:20_1"/>
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">R1</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] ) )
 <b>by </b><i/>;<br/>


</div>
