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

<b>consider </b><font color="Maroon">a'</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C1</font>, <font color="Maroon">b'</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C2</font>;<br/>
<a NAME="E2:148_1_1"/>
<font color="Maroon">a'</font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Maroon">b'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font> <a href="cohsp_1.html#K15" target="_self">"/\"</a> <font color="Maroon">C2</font>
 
;<br/>
<b>hence </b><a NAME="E3:148_1_1"/>
not <font color="Maroon">C1</font> <a href="cohsp_1.html#K15" target="_self">"/\"</a> <font color="Maroon">C2</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="classes1.html#D1">CLASSES1:def 1</a>
<div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:148_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font> <a href="cohsp_1.html#K15" target="_self">"/\"</a> <font color="Maroon">C2</font>
 ;<br/><b>then consider </b><font color="Maroon">aa</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C1</font>, <font color="Maroon">bb</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C2</font><b> such that </b><br/><a NAME="E3:148_1_1_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">aa</font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Maroon">bb</font>
 ;<br/><b>assume </b><a NAME="E4:148_1_1_1"/>
<font color="Maroon">b</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font>
 ;<br/><b>then consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:148_1_1_1"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Maroon">y1</font> &amp; <font color="Maroon">x1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">aa</font> &amp; <font color="Maroon">y1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">bb</font> )
 <b>by </b><i><a class="ref" href="cohsp_1.html#T23" target="_self">Th23</a>, <a class="ref" href="cohsp_1.html#T21" target="_self">Th21</a></i>;<br/><a NAME="E7:148_1_1_1"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font> &amp; <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C2</font> )
 <b>by </b><i>, <a class="ref" href="classes1.html#D1">CLASSES1:def 1</a></i>;<br/><b>hence </b><a NAME="E8:148_1_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font> <a href="cohsp_1.html#K15" target="_self">"/\"</a> <font color="Maroon">C2</font>
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="cohsp_1.html#D1" target="_self">COHSP_1:def 1</a><br/>

<b>assume </b><a NAME="E5:148_1_1"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>   st <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> holds <br/><font color="Olive">a</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font> <a href="cohsp_1.html#K15" target="_self">"/\"</a> <font color="Maroon">C2</font>
 ;<br/>

<b>set </b><font color="Maroon">A1</font> = <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C1</font> :  ex <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C2</font> st <font color="Olive">a</font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon">A2</font> = <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C2</font> :  ex <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C1</font> st <font color="Olive">a</font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="148_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:148_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C1</font> :  ex <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C2</font> st <font color="Olive">a</font> <a href="cohsp_1.html#K14" target="_self">U+</a> <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">ax</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C1</font><b> such that </b><br/><a NAME="E3:148_1_1_2"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ax</font> &amp;  ex <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C2</font> st <font color="Maroon">ax</font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/><b>consider </b><font color="Maroon">bx</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C2</font><b> such that </b><br/><a NAME="E5:148_1_1_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">ax</font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Maroon">bx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/><b>assume </b><a NAME="E6:148_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C1</font> :  ex <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C2</font> st <font color="Olive">a</font> <a href="cohsp_1.html#K14" target="_self">U+</a> <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">ay</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C1</font><b> such that </b><br/><a NAME="E8:148_1_1_2"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ay</font> &amp;  ex <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C2</font> st <font color="Maroon">ay</font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/><b>consider </b><font color="Maroon">by1</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C2</font><b> such that </b><br/><a NAME="E10:148_1_1_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">ay</font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Maroon">by1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/><a NAME="E11:148_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">ax</font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Maroon">bx</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">ay</font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Maroon">by1</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font> <a href="cohsp_1.html#K15" target="_self">"/\"</a> <font color="Maroon">C2</font>
 <b>by </b><i><a class="ref" href="cohsp_1.html#T23" target="_self">Th23</a>, , </i>;<br/><a NAME="E12:148_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">ax</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ay</font></span>)</span> <a href="cohsp_1.html#K14" target="_self">U+</a> <span class="p1">(<span class="default"><font color="Maroon">bx</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">by1</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font> <a href="cohsp_1.html#K15" target="_self">"/\"</a> <font color="Maroon">C2</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E13:148_1_1_2"/>
<font color="Maroon">x</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<a NAME="E7:148_1_1"/><b>then </b><i><font color="Green">E45</font></i>: 
 <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C1</font> :  ex <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C2</font> st <font color="Olive">a</font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span>  <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="148_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:148_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C2</font> :  ex <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C1</font> st <font color="Olive">a</font> <a href="cohsp_1.html#K14" target="_self">U+</a> <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">ax</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C2</font><b> such that </b><br/><a NAME="E3:148_1_1_3"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ax</font> &amp;  ex <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C1</font> st <font color="Olive">b</font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Maroon">ax</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/><b>consider </b><font color="Maroon">bx</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C1</font><b> such that </b><br/><a NAME="E5:148_1_1_3"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">bx</font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Maroon">ax</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/><b>assume </b><a NAME="E6:148_1_1_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C2</font> :  ex <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C1</font> st <font color="Olive">a</font> <a href="cohsp_1.html#K14" target="_self">U+</a> <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">ay</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C2</font><b> such that </b><br/><a NAME="E8:148_1_1_3"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ay</font> &amp;  ex <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C1</font> st <font color="Olive">b</font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Maroon">ay</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/><b>consider </b><font color="Maroon">by1</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C1</font><b> such that </b><br/><a NAME="E10:148_1_1_3"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">by1</font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Maroon">ay</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/><a NAME="E11:148_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">bx</font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Maroon">ax</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">by1</font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Maroon">ay</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font> <a href="cohsp_1.html#K15" target="_self">"/\"</a> <font color="Maroon">C2</font>
 <b>by </b><i><a class="ref" href="cohsp_1.html#T23" target="_self">Th23</a>, , <a class="ref" href="cohsp_1.html#T24" target="_self">Th24</a></i>;<br/><a NAME="E12:148_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">bx</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">by1</font></span>)</span> <a href="cohsp_1.html#K14" target="_self">U+</a> <span class="p1">(<span class="default"><font color="Maroon">ax</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ay</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font> <a href="cohsp_1.html#K15" target="_self">"/\"</a> <font color="Maroon">C2</font>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E10:148_1_1_3"><i><font color="Green">E80</font></i></a></i>;<br/><b>hence </b><a NAME="E13:148_1_1_3"/>
<font color="Maroon">x</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C2</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<a NAME="E9:148_1_1"/><b>then </b><i><font color="Green">E81</font></i>: 
 <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C2</font> :  ex <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C1</font> st <font color="Olive">a</font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span>  <a href="hidden.html#R2">in</a> <font color="Maroon">C2</font>
 
<b>by </b><i/>;<br/>
<a NAME="E10:148_1_1"/>
<span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C1</font> :  ex <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C2</font> st <font color="Olive">a</font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span> </span>)</span> <a href="cohsp_1.html#K14" target="_self">U+</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C2</font> :  ex <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C1</font> st <font color="Olive">a</font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="148_1_1_4"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>,<a class="ref" href="tarski.html#D3">TARSKI:def 3</a>
<div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:148_1_1_4_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C1</font> :  ex <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C2</font> st <font color="Olive">a</font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span> </span>)</span> <a href="cohsp_1.html#K14" target="_self">U+</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C2</font> :  ex <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C1</font> st <font color="Olive">a</font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span> </span>)</span>
 ;<br/><a NAME="E2:148_1_1_4_1"/><b>then </b><i><font color="Green">E84</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; ( ( <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C1</font> :  ex <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C2</font> st <font color="Olive">a</font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span>  ) or ( <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C2</font> :  ex <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C1</font> st <font color="Olive">a</font> <a href="cohsp_1.html#K14" target="_self">U+</a> <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="cohsp_1.html#D14" target="_self">Def14</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="148_1_1_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:148_1_1_4_1_1"/>
( ( <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C1</font> :  ex <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C2</font> st <font color="Olive">a</font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span>  ) or ( <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C2</font> :  ex <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C1</font> st <font color="Olive">a</font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span>  ) )
 </span><b>by </b><i>, <a class="ref" href="cohsp_1.html#D14" target="_self">Def14</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="148_1_1_4_1_1_1"><b>suppose </b></a><a NAME="E1:148_1_1_4_1_1_1"/><i><font color="Green">E85</font></i>: 
( <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C1</font> :  ex <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C2</font> st <font color="Olive">a</font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span>  )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:148_1_1_4_1_1_1"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C1</font> :  ex <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C2</font> st <font color="Olive">a</font> <a href="cohsp_1.html#K14" target="_self">U+</a> <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">ax</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C1</font><b> such that </b><br/><a NAME="E5:148_1_1_4_1_1_1"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ax</font> &amp;  ex <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C2</font> st <font color="Maroon">ax</font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="cohsp_1.html#D17" target="_self">Def17</a></i>;<br/><b>consider </b><font color="Maroon">bx</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C2</font><b> such that </b><br/><a NAME="E7:148_1_1_4_1_1_1"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">ax</font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Maroon">bx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="cohsp_1.html#T25" target="_self">Th25</a></i>;<br/><a NAME="E8:148_1_1_4_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ax</font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Maroon">bx</font>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E5:148_1_1_4_1_1_1"><i><font color="Green">E91</font></i></a>, <a class="ref" href="cohsp_1.html#T24" target="_self">Th24</a>, <a class="ref" href="cohsp_1.html#D17" target="_self">Def17</a>, <a class="ref" href="cohsp_1.html#T25" target="_self">Th25</a>, <a class="ref" href="cohsp_1.html#D15" target="_self">Def15</a></i>;<br/><b>hence </b><a NAME="E9:148_1_1_4_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">A</font>
 <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="148_1_1_4_1_1_2"><b>suppose </b></a><a NAME="E1:148_1_1_4_1_1_2"/><i><font color="Green">E119</font></i>: 
( <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C2</font> :  ex <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C1</font> st <font color="Olive">a</font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span>  )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:148_1_1_4_1_1_2"/><i><font color="Green">E120</font></i>: 
( <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C2</font> :  ex <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C1</font> st <font color="Olive">a</font> <a href="cohsp_1.html#K14" target="_self">U+</a> <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">bx</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C2</font><b> such that </b><br/><a NAME="E5:148_1_1_4_1_1_2"/><i><font color="Green">E121</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">bx</font> &amp;  ex <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C1</font> st <font color="Olive">a</font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Maroon">bx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="cohsp_1.html#D17" target="_self">Def17</a></i>;<br/><b>consider </b><font color="Maroon">ax</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C1</font><b> such that </b><br/><a NAME="E7:148_1_1_4_1_1_2"/><i><font color="Green">E181</font></i>: 
<font color="Maroon">ax</font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Maroon">bx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="cohsp_1.html#T25" target="_self">Th25</a></i>;<br/><a NAME="E8:148_1_1_4_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ax</font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Maroon">bx</font>
 <b>by </b><i>, <a class="ref" href="cohsp_1.html#T24" target="_self">Th24</a>, <a class="ref" href="cohsp_1.html#D17" target="_self">Def17</a>, <a class="ref" href="cohsp_1.html#T25" target="_self">Th25</a>, <a class="ref" href="cohsp_1.html#D16" target="_self">Def16</a></i>;<br/><b>hence </b><a NAME="E9:148_1_1_4_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">A</font>
 <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>

<b>let </b><font color="Maroon">x</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="E2:148_1_1_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">A</font>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:148_1_1_4"/><i><font color="Green">E182</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E5:148_1_1_4"/>
<font color="Maroon">a</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font> <a href="cohsp_1.html#K15" target="_self">"/\"</a> <font color="Maroon">C2</font>
 
<b>by </b><i><a class="ref" href="cohsp_1.html#T23" target="_self">Th23</a>, </i>;<br/>
<b>then consider </b><font color="Maroon">aa</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C1</font>, <font color="Maroon">bb</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C2</font><b> such that </b><br/><a NAME="E7:148_1_1_4"/><i><font color="Green">E183</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">aa</font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Maroon">bb</font>
 ;<br/>
<a NAME="E8:148_1_1_4"/>
( <font color="Maroon">aa</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C1</font> :  ex <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C2</font> st <font color="Olive">a</font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span>  &amp; <font color="Maroon">bb</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C2</font> :  ex <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C1</font> st <font color="Olive">a</font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span>  )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:148_1_1_4"/><b>then </b>
( <font color="Maroon">aa</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C1</font> :  ex <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C2</font> st <font color="Olive">a</font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span>  &amp; <font color="Maroon">bb</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C2</font> :  ex <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C1</font> st <font color="Olive">a</font> <a href="cohsp_1.html#K14" target="_self">U+</a> <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#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E10:148_1_1_4"/><b>then </b>
<font color="Maroon">aa</font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Maroon">bb</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C1</font> :  ex <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C2</font> st <font color="Olive">a</font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span> </span>)</span> <a href="cohsp_1.html#K14" target="_self">U+</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C2</font> :  ex <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C1</font> st <font color="Olive">a</font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span> </span>)</span>
 
<b>by </b><i><a class="ref" href="cohsp_1.html#T20" target="_self">Th20</a></i>;<br/>
<b>hence </b><a NAME="E11:148_1_1_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C1</font> :  ex <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C2</font> st <font color="Olive">a</font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span> </span>)</span> <a href="cohsp_1.html#K14" target="_self">U+</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C2</font> :  ex <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C1</font> st <font color="Olive">a</font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span> </span>)</span>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:148_1_1"/>
 <a href="tarski.html#K3">union</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font> <a href="cohsp_1.html#K15" target="_self">"/\"</a> <font color="Maroon">C2</font>
 <b>by </b><i>, </i>;<br/>


</div>
