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

<b>set </b><font color="Maroon" title="c2">irr</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b1">z</font> <a href="armstrng.html#R1" title="ARMSTRNG:pred.1">is_/\-irreducible_in</a> <font color="Maroon" title="c1">X</font> </span>}</span> ;<br/>
<a NAME="E1:9_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b1">z</font> <a href="armstrng.html#R1" title="ARMSTRNG:pred.1">is_/\-irreducible_in</a> <font color="Maroon" title="c1">X</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:9_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b1">z</font> <a href="armstrng.html#R1" title="ARMSTRNG:pred.1">is_/\-irreducible_in</a> <font color="Maroon" title="c1">X</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">z</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:9_1_1_1"/><i><font color="Green" title="E6">A1</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">z</font> &amp; <font color="Maroon" title="c4">z</font> <a href="armstrng.html#R1" title="ARMSTRNG:pred.1">is_/\-irreducible_in</a> <font color="Maroon" title="c1">X</font> )
 ;<br/>
<b>thus </b><a NAME="E4:9_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E3:9_1_1_1"><i><font color="Green" title="E6">A1</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c3">irr</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b1">z</font> <a href="armstrng.html#R1" title="ARMSTRNG:pred.1">is_/\-irreducible_in</a> <font color="Maroon" title="c1">X</font> </span>}</span>  as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c3">irr</font>
;<br/>

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:9_1_1_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">irr</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c5">z</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:9_1_1_2"/><i><font color="Green" title="E6">A2</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c5">z</font> <a href="armstrng.html#R1" title="ARMSTRNG:pred.1">is_/\-irreducible_in</a> <font color="Maroon" title="c1">X</font> )
 ;<br/><b>thus </b><a NAME="E4:9_1_1_2"/>
<font color="Maroon" title="c4">x</font> <a href="armstrng.html#R1" title="ARMSTRNG:pred.1">is_/\-irreducible_in</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E3:9_1_1_2"><i><font color="Green" title="E6">A2</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E4:9_1_1"/><i><font color="Green" title="E6">A3</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="armstrng.html#R1" title="ARMSTRNG:pred.1">is_/\-irreducible_in</a> <font color="Maroon" title="c1">X</font>
 ;<br/>

<a NAME="E5:9_1_1"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="armstrng.html#D2" target="_self" title="ARMSTRNG:def.2">Def2</a></i>;<br/>
<b>hence </b><a NAME="E6:9_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">irr</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E4:9_1_1"><i><font color="Green" title="E6">A3</font></i></a></i>;<br/>


</div>
