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

<b>set </b><font color="Maroon">X</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>;<br/>
<a NAME="E1:37_1_1"/>
 <a href="pua2mss1.html#K6" target="_self">DomRel</a> <font color="Maroon">A</font> <a href="relat_2.html#R1">is_reflexive_in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D1">RELAT_2:def 1</a><br/>

<a NAME="E1:37_1_1_1"/>
for <font color="Olive">f</font> being  <a href="pua2mss1.html#NM3">operation</a> of <font color="Maroon">A</font><br/> for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="finseq_1.html#NM1">FinSequence</a> holds <br/> ( <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Olive">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Olive">f</font> iff <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Olive">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Olive">f</font> )
 
;<br/>
<b>hence </b><a NAME="E2:37_1_1_1"/>
(  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> or <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>  <a href="pua2mss1.html#K6" target="_self">DomRel</a> <font color="Maroon">A</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:37_1_1"/><b>then </b><i><font color="Green">E29</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K6" target="_self">DomRel</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> &amp;  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K6" target="_self">DomRel</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> )
 
<b>by </b><i><a class="ref" href="orders_1.html#T98">ORDERS_1:98</a></i>;<br/>
<b>hence </b><a NAME="E3:37_1_1"/>
 <a href="pua2mss1.html#K6" target="_self">DomRel</a> <font color="Maroon">A</font> is <a href="partfun1.html#V1">total</a>
 <b>by </b><i><a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a></i>;<br/>

<a NAME="E4:37_1_1"/>
 <a href="pua2mss1.html#K6" target="_self">DomRel</a> <font color="Maroon">A</font> <a href="relat_2.html#R3">is_symmetric_in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D3">RELAT_2:def 3</a><br/>

<b>assume </b><a NAME="E1:37_1_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> ;<br/>
<b>assume </b><a NAME="E3:37_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="pua2mss1.html#K6" target="_self">DomRel</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E4:37_1_1_2"/><b>then </b>
for <font color="Olive">f</font> being  <a href="pua2mss1.html#NM3">operation</a> of <font color="Maroon">A</font><br/> for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="finseq_1.html#NM1">FinSequence</a> holds <br/> ( <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x'</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Olive">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Olive">f</font> iff <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">y'</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Olive">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Olive">f</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:37_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="pua2mss1.html#K6" target="_self">DomRel</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:37_1_1"/>
 <a href="pua2mss1.html#K6" target="_self">DomRel</a> <font color="Maroon">A</font> is <a href="relat_2.html#V3">symmetric</a>
 <b>by </b><i>, <a class="ref" href="relat_2.html#D11">RELAT_2:def 11</a></i>;<br/>

<a NAME="E6:37_1_1"/>
 <a href="pua2mss1.html#K6" target="_self">DomRel</a> <font color="Maroon">A</font> <a href="relat_2.html#R8">is_transitive_in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D8">RELAT_2:def 8</a><br/>

<b>assume </b><a NAME="E1:37_1_1_3"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font>, <font color="Maroon">z'</font> = <font color="Maroon">z</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> ;<br/>
<b>assume </b><a NAME="E3:37_1_1_3"/><i><font color="Green">E36</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="pua2mss1.html#K6" target="_self">DomRel</a> <font color="Maroon">A</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="pua2mss1.html#K6" target="_self">DomRel</a> <font color="Maroon">A</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">f</font> be   <a href="pua2mss1.html#NM3">operation</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="finseq_1.html#NM1">FinSequence</a>, <font color="Maroon">b</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/><a NAME="E1:37_1_1_3_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x'</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> iff <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">y'</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E2:37_1_1_3_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x'</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> iff <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">z'</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:37_1_1_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="pua2mss1.html#K6" target="_self">DomRel</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:37_1_1"/>
 <a href="pua2mss1.html#K6" target="_self">DomRel</a> <font color="Maroon">A</font> is <a href="relat_2.html#V8">transitive</a>
 <b>by </b><i>, <a class="ref" href="relat_2.html#D16">RELAT_2:def 16</a></i>;<br/>


</div>
