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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>] <b>means </b>( <font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>1</sub></font> is <a href="waybel_3.html#V1">compact</a> );<br/>
<b>consider </b><font color="Maroon">X</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E2:6_1_1"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds <br/> ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">y</font>] )
 <b>from </b><i><a class="ref" href="subset_1.html#S3">SUBSET_1:sch 3</a>();<br/></i>
<b>set </b><font color="Maroon">Z</font> = <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : ( <font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">y</font> &amp; <font color="Olive">y</font> is <a href="waybel_3.html#V1">compact</a> ) </span>}</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E1:6_1_1_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> implies <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : ( <font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">y</font> &amp; <font color="Olive">y</font> is <a href="waybel_3.html#V1">compact</a> ) </span>}</span>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_1_1_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">z1</font> = <font color="Maroon">z</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E3:6_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">z1</font> &amp; <font color="Maroon">z1</font> is <a href="waybel_3.html#V1">compact</a> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:6_1_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : ( <font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">y</font> &amp; <font color="Olive">y</font> is <a href="waybel_3.html#V1">compact</a> ) </span>}</span> 
 ;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E2:6_1_1_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : ( <font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">y</font> &amp; <font color="Olive">y</font> is <a href="waybel_3.html#V1">compact</a> ) </span>}</span>  implies <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_1_1_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : ( <font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">y</font> &amp; <font color="Olive">y</font> is <a href="waybel_3.html#V1">compact</a> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">v</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:6_1_1_1_2"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">v</font> &amp; <font color="Maroon">v</font> is <a href="waybel_3.html#V1">compact</a> )
 ;<br/>
<b>thus </b><a NAME="E4:6_1_1_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:6_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : ( <font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">y</font> &amp; <font color="Olive">y</font> is <a href="waybel_3.html#V1">compact</a> ) </span>}</span>  is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


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