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

<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">R</font> be   <a href="relset_1.html#NM3" title="RELSET_1:NM.3">Relation</a> of <font color="Maroon" title="c1">X</font>;<br/>



<b>assume </b><a NAME="E1:9"/><i><font color="Green" title="E8">A1</font></i>: 
<font color="Maroon" title="c2">R</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a> <font color="Maroon" title="c1">X</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font>, <font color="Maroon" title="c5">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:9_1"/><i><font color="Green" title="E9">A2</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> <a href="finseq_1.html#K18" title="FINSEQ_1:func.18">[*]</a>  &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> <a href="finseq_1.html#K18" title="FINSEQ_1:func.18">[*]</a>  )
 ;<br/><a NAME="E2:9_1"/><b>then </b><i><font color="Green" title="E10">A3</font></i>: 
<font color="Maroon" title="c2">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>
 <b>by </b><i><a class="ref" href="rewrite1.html#T21" title="REWRITE1:th.21">REWRITE1:21</a></i>;<br/><a NAME="E3:9_1"/>
<font color="Maroon" title="c2">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font>
 <b>by </b><i><a class="txt" href="taxonom1.html#E1:9_1"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="rewrite1.html#T21" title="REWRITE1:th.21">REWRITE1:21</a></i>;<br/><b>hence </b><a NAME="E4:9_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> <a href="finseq_1.html#K18" title="FINSEQ_1:func.18">[*]</a> 
 <b>by </b><i><a class="txt" href="taxonom1.html#E1:9"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="taxonom1.html#E1:9_1"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="taxonom1.html#E2:9_1"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="taxonom1.html#T6" target="_self" title="TAXONOM1:th.6">Th6</a>, <a class="ref" href="rewrite1.html#T17" title="REWRITE1:th.17">REWRITE1:17</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:9"/>
<font color="Maroon" title="c2">R</font> <a href="finseq_1.html#K18" title="FINSEQ_1:func.18">[*]</a>  <a href="relat_2.html#R8" title="RELAT_2:pred.8">is_transitive_in</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="relat_2.html#D8" title="RELAT_2:def.8">RELAT_2:def 8</a></i>;<br/>


</div>
