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

<b>thus </b><a NAME="E1:12_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="lfuzzy_1.html#V1" target="_self">reflexive</a> iff for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>2</sub></font> <a href="lfuzzy_0.html#K7">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> 1 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_1"><b>proof </b></a><div class="add">

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:12_1_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="lfuzzy_1.html#V1" target="_self">reflexive</a>
 ;<br/><a NAME="E2:12_1_1_1_1"/><b>then </b><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="fuzzy_1.html#NR2" target="_self">c=</a> 
 <b>by </b><i><a class="ref" href="lfuzzy_1.html#D2" target="_self">Def2</a></i>;<br/><b>thus </b><a NAME="E3:12_1_1_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>2</sub></font> <a href="lfuzzy_0.html#K7">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E1:12_1_1_1_1_1"/>
( <span class="p1">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lfuzzy_0.html#K7">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K1">]</a></span> &amp; <span class="p1">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="txt" href="lfuzzy_1.html#E2:12_1_1_1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="lfuzzy_1.html#D1" target="_self">Def1</a>, <a class="ref" href="fuzzy_4.html#D4">FUZZY_4:def 4</a></i>;<br/>
<a NAME="E2:12_1_1_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="lfuzzy_0.html#K7">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="lfuzzy_0.html#K7">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 )
 
<b>by </b><i><a class="ref" href="fuzzy_4.html#T4">FUZZY_4:4</a></i>;<br/>
<b>hence </b><a NAME="E3:12_1_1_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="lfuzzy_0.html#K7">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>


</div><b>end;</b></div></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:12_1_1_1"/><i><font color="Green">E9</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>2</sub></font> <a href="lfuzzy_0.html#K7">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> 1
 ;<br/>

<a NAME="E3:12_1_1_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <span class="p1">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lfuzzy_0.html#K7">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_1_1_1_2_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> or  not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:12_1_1_1_2_1_1"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><a NAME="E2:12_1_1_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="fuzzy_4.html#D4">FUZZY_4:def 4</a></i>;<br/><b>hence </b><a NAME="E3:12_1_1_1_2_1_1"/>
<span class="p1">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lfuzzy_0.html#K7">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i><a class="txt" href="lfuzzy_1.html#E2:12_1_1_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="lfuzzy_1.html#E1:12_1_1_1_2_1_1"><i><font color="Green">E10</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:12_1_1_1_2_1_2"/>
not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><a NAME="E2:12_1_1_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="fuzzy_4.html#D4">FUZZY_4:def 4</a></i>;<br/><b>hence </b><a NAME="E3:12_1_1_1_2_1_2"/>
<span class="p1">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lfuzzy_0.html#K7">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i><a class="ref" href="fuzzy_4.html#T4">FUZZY_4:4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E4:12_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="fuzzy_1.html#NR2" target="_self">c=</a> 
 
<b>by </b><i><a class="ref" href="lfuzzy_1.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E5:12_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="lfuzzy_1.html#V1" target="_self">reflexive</a>
 <b>by </b><i><a class="ref" href="lfuzzy_1.html#D2" target="_self">Def2</a></i>;<br/>


</div><b>end;</b></div>


</div>
