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

<b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <a href="orders_2.html#G1">RelStr</a>(# <font color="Olive">A</font>,<font color="Olive">B</font> #) where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>, <font color="Olive">B</font> is    <a href="relset_1.html#M2">Relation</a> of <font color="Olive">A</font>,<font color="Olive">A</font> : <font color="Olive">B</font> <a href="tarski.html#R1">c=</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">L</font> </span>}</span> ;<br/>
<b>consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/>
<a NAME="E2:30_1_1"/>
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">a</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">L</font> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#D13">YELLOW_0:def 13</a>, <a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<a NAME="E3:30_1_1"/><b>then </b>
 <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <a href="orders_2.html#G1">RelStr</a>(# <font color="Olive">A</font>,<font color="Olive">B</font> #) where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>, <font color="Olive">B</font> is    <a href="relset_1.html#M2">Relation</a> of <font color="Olive">A</font>,<font color="Olive">A</font> : <font color="Olive">B</font> <a href="tarski.html#R1">c=</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">L</font> </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <a href="orders_2.html#G1">RelStr</a>(# <font color="Olive">A</font>,<font color="Olive">B</font> #) where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>, <font color="Olive">B</font> is    <a href="relset_1.html#M2">Relation</a> of <font color="Olive">A</font>,<font color="Olive">A</font> : <font color="Olive">B</font> <a href="tarski.html#R1">c=</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">L</font> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">R</font> being   <a href="orders_2.html#L1">RelStr</a>  st <br/>( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">R</font> &amp; <font color="Maroon">a<sub>1</sub></font> is    <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Olive">R</font> );<br/>
<b>consider </b><font color="Maroon">S</font> being  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a> <b> such that </b><br/><a NAME="E6:30_1_1"/><i><font color="Green">E31</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E7:30_1_1"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> holds <br/> ( <font color="Olive">a</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>,<font color="Olive">b</font>] )
 <b>from </b><i><a class="ref" href="yellow_0.html#S1">YELLOW_0:sch 1</a>();<br/></i>
<b>take </b>
<font color="Maroon">S</font>
;<br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:30_1_1_1_1"/>
<font color="Maroon">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>
 ;<br/><a NAME="E2:30_1_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/><a NAME="E3:30_1_1_1_1"/><b>then </b>
 ex <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> ex <font color="Olive">B</font> being   <a href="relset_1.html#M2">Relation</a> of <font color="Olive">A</font>,<font color="Olive">A</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# <font color="Olive">A</font>,<font color="Olive">B</font> #) &amp; <font color="Olive">B</font> <a href="tarski.html#R1">c=</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">L</font> )
 ;<br/><b>hence </b><a NAME="E4:30_1_1_1_1"/>
<font color="Maroon">x</font> is  <a href="orders_2.html#V1">strict</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#D13">YELLOW_0:def 13</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:30_1_1_1"/>
<font color="Maroon">x</font> is  <a href="orders_2.html#V1">strict</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">L</font>
 ;<br/><b>then reconsider </b><font color="Maroon">R</font> = <font color="Maroon">x</font> as  <a href="orders_2.html#V1">strict</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">L</font> ;<br/><a NAME="E4:30_1_1_1"/>
( <font color="Maroon">R</font> <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> #) &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> &amp; the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="tarski.html#R1">c=</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">L</font> )
 <b>by </b><i><a class="ref" href="yellow_0.html#D13">YELLOW_0:def 13</a></i>;<br/><a NAME="E5:30_1_1_1"/><b>then </b>
<font color="Maroon">R</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><b>hence </b><a NAME="E6:30_1_1_1"/>
<font color="Maroon">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E9:30_1_1"/>
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> holds <br/> ( <font color="Olive">a</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b</font> iff  ex <font color="Olive">R</font> being   <a href="orders_2.html#L1">RelStr</a>  st <br/>( <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">R</font> &amp; <font color="Olive">a</font> is    <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Olive">R</font> ) )
 <b>by </b><i/>;<br/>


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