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

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

<div><i><font color="Green">E30</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_1"/>
<font color="Maroon">R</font> is <a href="relat_2.html#V8" target="_self">transitive</a>
 ;<br/><a NAME="E2:28_1"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">R</font> <a href="relat_2.html#R8" target="_self">is_transitive_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="relat_2.html#D3" target="_self">Def3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:28_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="relat_1.html#K5">*</a> <font color="Maroon">R</font>
 ;<br/><b>then consider </b><font color="Maroon">c</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:28_1_1"/><i><font color="Green">E32</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#D8">RELAT_1:def 8</a></i>;<br/><a NAME="E4:28_1_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> )
 <b>by </b><i><a class="ref" href="relat_2.html#D10" target="_self">Def10</a>, <a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/><b>hence </b><a NAME="E5:28_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="relat_2.html#D9" target="_self">Def9</a>, <a class="ref" href="relat_2.html#D10" target="_self">Def10</a>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:28_1"/>
<font color="Maroon">R</font> <a href="relat_1.html#K5">*</a> <font color="Maroon">R</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D3">RELAT_1:def 3</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">R</font> <a href="relat_1.html#K5">*</a> <font color="Maroon">R</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:28_2_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 ;<br/><a NAME="E2:28_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="relat_1.html#K5">*</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D8">RELAT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E3:28_2_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="relat_2.html#D9" target="_self">Def9</a></i>;<br/></div><b>end;</b></div><a NAME="E3:28_2"/><b>then </b>
<font color="Maroon">R</font> <a href="relat_2.html#R8" target="_self">is_transitive_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:28_2"/>
<font color="Maroon">R</font> is <a href="relat_2.html#V8" target="_self">transitive</a>
 <b>by </b><i><a class="ref" href="relat_2.html#D3" target="_self">Def3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:28"/>
( <font color="Maroon">R</font> is <a href="relat_2.html#V8" target="_self">transitive</a> iff <font color="Maroon">R</font> <a href="relat_1.html#K5">*</a> <font color="Maroon">R</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font> )
 <b>by </b><i><a class="ref" href="relat_2.html#D8" target="_self">Def8</a></i>;<br/>


</div>
