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

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_1_1_1"/>
( <font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a> or  not <font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_1_1"><b>suppose </b></a><a NAME="E1:52_1_1_1_1"/>
<font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:52_1_1_1_1"/>
<font color="Maroon">Y</font> <a href="yellow_1.html#K6" target="_self">|^</a> <font color="Maroon">X</font> is <a href="orders_2.html#V2">reflexive</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_1_2"><b>suppose </b></a><a NAME="E1:52_1_1_1_2"/>
not <font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">X</font> = <font color="Maroon">X</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/><a NAME="E3:52_1_1_1_2"/>
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="yellow_1.html#K6" target="_self">|^</a> <font color="Maroon">X</font></span>)</span> holds  <font color="Olive">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_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 <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="yellow_1.html#K6" target="_self">|^</a> <font color="Maroon">X</font></span>)</span>;<br/>

<a NAME="E1:52_1_1_1_2_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" target="_self">product</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">Y</font></span>)</span></span>)</span>
 
;<br/>
<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" target="_self">product</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">Y</font></span>)</span></span>)</span> ;<br/>
<a NAME="E3:52_1_1_1_2_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">Y</font></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> <b>by </b><i/>;<br/>
<a NAME="E5:52_1_1_1_2_1"/>
 ex <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> &amp; <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> &amp; ( for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/> ex <font color="Olive">R</font> being   <a href="orders_2.html#L1">RelStr</a>  ex <font color="Olive">xi</font>, <font color="Olive">yi</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Olive">R</font> st <br/>( <font color="Olive">R</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">Y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">xi</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">yi</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">xi</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">yi</font> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">x1</font>
;<br/>

<b>take </b>
<font color="Maroon">x1</font>
;<br/>

<b>thus </b><a NAME="E1:52_1_1_1_2_1_1"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> )
 ;<br/>

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E2:52_1_1_1_2_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> ;<br/>
<b>take </b><font color="Maroon">R</font> = <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">Y</font></span>)</span> <a href="yellow_1.html#K4" target="_self">.</a> <font color="Maroon">i</font>;<br/>

<b>reconsider </b><font color="Maroon">xi</font> = <font color="Maroon">x1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font>, <font color="Maroon">yi</font> = <font color="Maroon">x1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> <b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<b>take </b>
<font color="Maroon">xi</font>
;<br/>

<b>take </b>
<font color="Maroon">yi</font>
;<br/>

<b>reconsider </b><font color="Maroon">xi1</font> = <font color="Maroon">xi</font>, <font color="Maroon">yi1</font> = <font color="Maroon">xi</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> ;<br/>
<a NAME="E6:52_1_1_1_2_1_1"/>
<font color="Maroon">xi1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">yi1</font>
 
;<br/>
<b>hence </b><a NAME="E7:52_1_1_1_2_1_1"/>
( <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">Y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">xi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">yi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">xi</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">yi</font> )
 <b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:52_1_1_1_2_1"/><b>then </b>
<font color="Maroon">x'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">x'</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:52_1_1_1_2_1"/><b>then </b>
<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> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" target="_self">product</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">Y</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E8:52_1_1_1_2_1"/><b>then </b>
<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> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="yellow_1.html#K6" target="_self">|^</a> <font color="Maroon">X</font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E9:52_1_1_1_2_1"/>
<font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:52_1_1_1_2"/>
<font color="Maroon">Y</font> <a href="yellow_1.html#K6" target="_self">|^</a> <font color="Maroon">X</font> is <a href="orders_2.html#V2">reflexive</a>
 <b>by </b><i><a class="ref" href="yellow_0.html#D1">YELLOW_0:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
