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

<b>let </b><font color="Maroon">R</font> be    <a href="orders_2.html#L1">RelStr</a> ;<br/>

<b>assume </b><a NAME="E1:14_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">R</font> is <a href="lfuzzy_0.html#V1" target="_self">real</a>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_1_1_1"/>
( <font color="Maroon">R</font> is <a href="struct_0.html#V3">empty</a> or  not <font color="Maroon">R</font> is <a href="struct_0.html#V3">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1_1"><b>suppose </b></a><a NAME="E1:14_1_1_1_1"/>
<font color="Maroon">R</font> is <a href="struct_0.html#V3">empty</a>
 ;<br/><div class="add"><a NAME="E2:14_1_1_1_1"/><b>then </b>
<font color="Maroon">R</font> is  <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> 
 ;<br/><b>hence </b><a NAME="E3:14_1_1_1_1"/>
( <font color="Maroon">R</font> is <a href="orders_2.html#V2">reflexive</a> &amp; <font color="Maroon">R</font> is <a href="orders_2.html#V4">antisymmetric</a> &amp; <font color="Maroon">R</font> is <a href="orders_2.html#V3">transitive</a> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1_2"><b>suppose </b></a><a NAME="E1:14_1_1_1_2"/>
not <font color="Maroon">R</font> is <a href="struct_0.html#V3">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">R'</font> = <font color="Maroon">R</font> as  non <a href="struct_0.html#V3">empty</a> <a href="lfuzzy_0.html#V1" target="_self">real</a>  <a href="orders_2.html#L1">RelStr</a>  <b>by </b><i/>;<br/><a NAME="E3:14_1_1_1_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">R'</font> is <a href="orders_2.html#V2">reflexive</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R'</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D1">YELLOW_0:def 1</a><br/>

<b>thus </b><a NAME="E1:14_1_1_1_2_1"/>
<font color="Maroon">x</font> <a href="lfuzzy_0.html#NR1" target="_self">&lt;&lt;=</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E4:14_1_1_1_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">R'</font> is <a href="orders_2.html#V4">antisymmetric</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R'</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R'</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D3">YELLOW_0:def 3</a><br/>

<b>assume </b><a NAME="E1:14_1_1_1_2_2"/>
( <font color="Maroon">x</font> <a href="lfuzzy_0.html#NR1" target="_self">&lt;&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="lfuzzy_0.html#NR2" target="_self">&gt;&gt;=</a> <font color="Maroon">y</font> )
 ;<br/>

<a NAME="E2:14_1_1_1_2_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="lfuzzy_0.html#NR5" target="_self">&lt;R=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="lfuzzy_0.html#NR7" target="_self">&gt;R=</a> <font color="Maroon">y</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:14_1_1_1_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:14_1_1_1_2"/>
<font color="Maroon">R'</font> is <a href="orders_2.html#V3">transitive</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R'</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R'</font>, <font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R'</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a><br/>

<b>assume </b><a NAME="E1:14_1_1_1_2_3"/>
( <font color="Maroon">x</font> <a href="lfuzzy_0.html#NR1" target="_self">&lt;&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="lfuzzy_0.html#NR1" target="_self">&lt;&lt;=</a> <font color="Maroon">z</font> )
 ;<br/>

<a NAME="E2:14_1_1_1_2_3"/><b>then </b>
( <font color="Maroon">x</font> <a href="lfuzzy_0.html#NR5" target="_self">&lt;R=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="lfuzzy_0.html#NR5" target="_self">&lt;R=</a> <font color="Maroon">z</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:14_1_1_1_2_3"/><b>then </b>
<font color="Maroon">x</font> <a href="lfuzzy_0.html#NR5" target="_self">&lt;R=</a> <font color="Maroon">z</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E4:14_1_1_1_2_3"/>
<font color="Maroon">x</font> <a href="lfuzzy_0.html#NR1" target="_self">&lt;&lt;=</a> <font color="Maroon">z</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E6:14_1_1_1_2"/>
( <font color="Maroon">R</font> is <a href="orders_2.html#V2">reflexive</a> &amp; <font color="Maroon">R</font> is <a href="orders_2.html#V4">antisymmetric</a> &amp; <font color="Maroon">R</font> is <a href="orders_2.html#V3">transitive</a> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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