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

<b>let </b><font color="Maroon">A</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:4"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">A</font> is <a href="ordinal1.html#V6">c=-linear</a>
 ;<br/>

<b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">a</font>,<font color="Olive">a</font></span><a href="zfmisc_1.html#K2">:]</a></span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> : <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon">Y</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="zfmisc_1.html#K2">:]</a></span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>, <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> : ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> ) </span>}</span> ;<br/>
<a NAME="E2:4"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">a</font>,<font color="Olive">a</font></span><a href="zfmisc_1.html#K2">:]</a></span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> : <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="zfmisc_1.html#K2">:]</a></span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>, <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> : ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_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="E1:4_1"/>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">a</font>,<font color="Olive">a</font></span><a href="zfmisc_1.html#K2">:]</a></span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> : <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E3:4_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">a</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/>
<b>thus </b><a NAME="E4:4_1"/>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="zfmisc_1.html#K2">:]</a></span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>, <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> : ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> ) </span>}</span> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E3:4"/><b>then </b><i><font color="Green">E39</font></i>: 
 <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">a</font>,<font color="Olive">a</font></span><a href="zfmisc_1.html#K2">:]</a></span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> : <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="zfmisc_1.html#K2">:]</a></span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>, <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> : ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T95">ZFMISC_1:95</a></i>;<br/>
<a NAME="E4:4"/>
 <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="zfmisc_1.html#K2">:]</a></span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>, <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> : ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">a</font>,<font color="Olive">a</font></span><a href="zfmisc_1.html#K2">:]</a></span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> : <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_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="E1:4_2"/>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="zfmisc_1.html#K2">:]</a></span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>, <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> : ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:4_2"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">z</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="zfmisc_1.html#K2">:]</a></span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>, <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> : ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">a</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>, <font color="Maroon">b</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E5:4_2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i/>;<br/>
<a NAME="E6:4_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="xboole_0.html#R3">are_c=-comparable</a> 
 
<b>by </b><i>, <a class="ref" href="ordinal1.html#D9">ORDINAL1:def 9</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_2_1"/>
( <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">b</font> or <font color="Maroon">b</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font> )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D9">XBOOLE_0:def 9</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_2_1_1"><b>suppose </b></a><a NAME="E1:4_2_1_1"/>
<font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><a NAME="E2:4_2_1_1"/><b>then </b>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">b</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T118">ZFMISC_1:118</a></i>;<br/><a NAME="E3:4_2_1_1"/><b>then </b>
( <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">b</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">b</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">a</font>,<font color="Olive">a</font></span><a href="zfmisc_1.html#K2">:]</a></span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> : <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span>  )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:4_2_1_1"/>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">a</font>,<font color="Olive">a</font></span><a href="zfmisc_1.html#K2">:]</a></span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> : <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_2_1_2"><b>suppose </b></a><a NAME="E1:4_2_1_2"/>
<font color="Maroon">b</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font>
 ;<br/><div class="add"><a NAME="E2:4_2_1_2"/><b>then </b>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">a</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T118">ZFMISC_1:118</a></i>;<br/><a NAME="E3:4_2_1_2"/><b>then </b>
( <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">a</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">a</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">a</font>,<font color="Olive">a</font></span><a href="zfmisc_1.html#K2">:]</a></span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> : <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span>  )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:4_2_1_2"/>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">a</font>,<font color="Olive">a</font></span><a href="zfmisc_1.html#K2">:]</a></span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> : <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E5:4"/><b>then </b>
 <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">a</font>,<font color="Olive">a</font></span><a href="zfmisc_1.html#K2">:]</a></span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> : <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="zfmisc_1.html#K2">:]</a></span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>, <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> : ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> ) </span>}</span> 
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E6:4"/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">a</font>,<font color="Olive">a</font></span><a href="zfmisc_1.html#K2">:]</a></span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> : <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span> 
 <b>by </b><i/>;<br/>


</div>
