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<b>let </b><font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">R</font> be   <a href="relset_1.html#NM3">Relation</a> of <font color="Maroon">A</font>, <font color="Maroon">E</font> be   <a href="relset_1.html#NM3">Relation</a> of <font color="Maroon">A</font>;<br/>



<b>assume </b><a NAME="E1:65"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> iff <font color="Olive">a</font>,<font color="Olive">b</font> <a href="rewrite1.html#R2">are_convertible_wrt</a> <font color="Maroon">R</font> )
 ;<br/>

<b>set </b><font color="Maroon">X</font> = <font color="Maroon">A</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><a NAME="E1:65_1"/>
<font color="Maroon">x</font>,<font color="Maroon">x</font> <a href="rewrite1.html#R2">are_convertible_wrt</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="rewrite1.html#T27">REWRITE1:27</a></i>;<br/><b>hence </b><a NAME="E2:65_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> implies <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">E</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E3:65"/><b>then </b>
<font color="Maroon">E</font> <a href="relat_2.html#R1">is_reflexive_in</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="relat_2.html#D1">RELAT_2:def 1</a></i>;<br/>
<a NAME="E4:65"/><b>then </b><i><font color="Green">E34</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">E</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp;  <a href="relat_1.html#K3">field</a> <font color="Maroon">E</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> )
 
<b>by </b><i><a class="ref" href="orders_1.html#T98">ORDERS_1:98</a></i>;<br/>
<b>hence </b><a NAME="E5:65"/>
<font color="Maroon">E</font> is <a href="partfun1.html#V1">total</a>
 <b>by </b><i><a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_2"><b>now </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> ;<br/><b>assume </b><a NAME="E1:65_2"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/><b>assume </b><a NAME="E2:65_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">E</font>
 ;<br/><a NAME="E3:65_2"/><b>then </b>
<font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="rewrite1.html#R2">are_convertible_wrt</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="txt" href="msualg_6.html#E1:65_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="msualg_6.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E4:65_2"/><b>then </b>
<font color="Maroon">y</font>,<font color="Maroon">x</font> <a href="rewrite1.html#R2">are_convertible_wrt</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="rewrite1.html#T32">REWRITE1:32</a></i>;<br/><b>hence </b><a NAME="E5:65_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">E</font>
 <b>by </b><i><a class="txt" href="msualg_6.html#E1:65_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="msualg_6.html#T3" target="_self">Th3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:65"/><b>then </b>
<font color="Maroon">E</font> <a href="relat_2.html#R3">is_symmetric_in</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="relat_2.html#D3">RELAT_2:def 3</a></i>;<br/>
<b>hence </b><a NAME="E8:65"/>
<font color="Maroon">E</font> is <a href="relat_2.html#V3">symmetric</a>
 <b>by </b><i><a class="ref" href="msualg_6.html#D1" target="_self">Def1</a>, <a class="ref" href="relat_2.html#D11">RELAT_2:def 11</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_3"><b>now </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> , <font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:65_3"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/><b>assume </b><a NAME="E2:65_3"/>
( <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">E</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">E</font> )
 ;<br/><a NAME="E3:65_3"/><b>then </b>
( <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="rewrite1.html#R2">are_convertible_wrt</a> <font color="Maroon">R</font> &amp; <font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="rewrite1.html#R2">are_convertible_wrt</a> <font color="Maroon">R</font> )
 <b>by </b><i>, <a class="ref" href="msualg_6.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E4:65_3"/><b>then </b>
<font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="rewrite1.html#R2">are_convertible_wrt</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="rewrite1.html#T31">REWRITE1:31</a></i>;<br/><b>hence </b><a NAME="E5:65_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">E</font>
 <b>by </b><i>, <a class="ref" href="msualg_6.html#T3" target="_self">Th3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E10:65"/><b>then </b>
<font color="Maroon">E</font> <a href="relat_2.html#R8">is_transitive_in</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="relat_2.html#D8">RELAT_2:def 8</a></i>;<br/>
<b>hence </b><a NAME="E11:65"/>
<font color="Maroon">E</font> is <a href="relat_2.html#V8">transitive</a>
 <b>by </b><i><a class="ref" href="msualg_6.html#D1" target="_self">Def1</a>, <a class="ref" href="relat_2.html#D16">RELAT_2:def 16</a></i>;<br/>


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