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

<b>let </b><font color="Maroon">L1</font> be  non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V6">finite</a> <a href="orders_2.html#V2">reflexive</a> <a href="orders_2.html#V3">transitive</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/>

<b>reconsider </b><font color="Maroon">Y</font> =  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L1</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="finset_1.html#T24">FINSET_1:24</a></i>;<br/>
<a NAME="E2:23"/><i><font color="Green">E40</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L1</font> : verum </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : <font color="Olive">X</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L1</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>proof </b></a><div class="add">

<b>let </b><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="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:23_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L1</font> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">X1</font> being   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L1</font><b> such that </b><br/><a NAME="E3:23_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X1</font>
 ;<br/>
<b>thus </b><a NAME="E4:23_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : <font color="Olive">X</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L1</font> </span>}</span> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E3:23"/><i><font color="Green">E43</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : <font color="Olive">X</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L1</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L1</font> : verum </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2"><b>proof </b></a><div class="add">

<b>let </b><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="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:23_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : <font color="Olive">X</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L1</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">X1</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E3:23_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X1</font>
 <b>and </b><br/><a NAME="E4:23_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">X1</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L1</font>
 ;<br/>
<b>thus </b><a NAME="E5:23_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L1</font> : verum </span>}</span> 
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<i><font color="Green">E49</font></i>:  <a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L1</font> = 
<span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L1</font> : verum </span>}</span> 
<b>by </b><i><a class="ref" href="waybel_0.html#D23">WAYBEL_0:def 23</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : <font color="Olive">X</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L1</font> </span>}</span> 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>
;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L1</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E5:23"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">X</font>) where <font color="Olive">X</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">X</font>] </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 
<b>from </b><i><a class="ref" href="pre_circ.html#S1">PRE_CIRC:sch 1</a>();<br/></i>
<b>hence </b><a NAME="E6:23"/>
 <a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L1</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i/>;<br/>


</div>
