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

<b>consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E2:148_2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="cohsp_1.html#K16" target="_self">"\/"</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="cohsp_1.html#T87" target="_self">Th87</a></i>;<br/>
<b>hence </b><a NAME="E3:148_2_1"/>
not <font color="Maroon">c<sub>1</sub></font> <a href="cohsp_1.html#K16" target="_self">"\/"</a> <font color="Maroon">c<sub>2</sub></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">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:148_2_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="cohsp_1.html#K16" target="_self">"\/"</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>7</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:148_2_1_1"/><i><font color="Green">E87</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Maroon">c<sub>7</sub></font> &amp; ( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 <b>by </b><i><a class="ref" href="cohsp_1.html#T88" target="_self">Th88</a></i>;<br/><b>assume </b><a NAME="E4:148_2_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:148_2_1_1"/><i><font color="Green">E88</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="cohsp_1.html#E3:148_2_1_1"><i><font color="Green">E87</font></i></a>, <a class="ref" href="cohsp_1.html#T80" target="_self">Th80</a></i>;<br/><a NAME="E7:148_2_1_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; ( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 <b>by </b><i><a class="txt" href="cohsp_1.html#E3:148_2_1_1"><i><font color="Green">E87</font></i></a>, <a class="txt" href="cohsp_1.html#E6:148_2_1_1"><i><font color="Green">E88</font></i></a>, <a class="ref" href="classes1.html#D1">CLASSES1:def 1</a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/><b>hence </b><a NAME="E8:148_2_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="cohsp_1.html#K16" target="_self">"\/"</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E6:148_2_1_1"><i><font color="Green">E88</font></i></a>, <a class="ref" href="cohsp_1.html#T87" target="_self">Th87</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>4</sub></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_2_1"/><i><font color="Green">E87</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="cohsp_1.html#K16" target="_self">"\/"</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="148_2_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:148_2_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:148_2_1_2"/><i><font color="Green">E88</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Maroon">c<sub>9</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/><b>consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E5:148_2_1_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E3:148_2_1_2"><i><font color="Green">E88</font></i></a></i>;<br/><b>assume </b><a NAME="E6:148_2_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E8:148_2_1_2"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Maroon">c<sub>11</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/><b>consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E10:148_2_1_2"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E8:148_2_1_2"><i><font color="Green">E90</font></i></a></i>;<br/><a NAME="E11:148_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="cohsp_1.html#K16" target="_self">"\/"</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E5:148_2_1"><i><font color="Green">E87</font></i></a>, <a class="txt" href="cohsp_1.html#E5:148_2_1_2"><i><font color="Green">E89</font></i></a>, <a class="txt" href="cohsp_1.html#E10:148_2_1_2"><i><font color="Green">E91</font></i></a></i>;<br/><a NAME="E12:148_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="cohsp_1.html#K14" target="_self">U+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="cohsp_1.html#K16" target="_self">"\/"</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="cohsp_1.html#T82" target="_self">Th82</a></i>;<br/><a NAME="E13:148_2_1_2"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>7</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="txt" href="cohsp_1.html#E3:148_2_1_2"><i><font color="Green">E88</font></i></a>, <a class="txt" href="cohsp_1.html#E8:148_2_1_2"><i><font color="Green">E90</font></i></a>, <a class="ref" href="cohsp_1.html#T87" target="_self">Th87</a></i>;<br/><b>hence </b><a NAME="E14:148_2_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="coh_sp.html#T1">COH_SP:1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:148_2_1"/><b>then </b><i><font color="Green">E88</font></i>: 
 <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="cohsp_1.html#D1" target="_self">Def1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="148_2_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:148_2_1_3"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:148_2_1_3"/><i><font color="Green">E89</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/><b>consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:148_2_1_3"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E3:148_2_1_3"><i><font color="Green">E89</font></i></a></i>;<br/><b>assume </b><a NAME="E6:148_2_1_3"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E8:148_2_1_3"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/><b>consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E10:148_2_1_3"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E8:148_2_1_3"><i><font color="Green">E91</font></i></a></i>;<br/><a NAME="E11:148_2_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="cohsp_1.html#K16" target="_self">"\/"</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E5:148_2_1"><i><font color="Green">E87</font></i></a>, <a class="txt" href="cohsp_1.html#E5:148_2_1_3"><i><font color="Green">E90</font></i></a>, <a class="txt" href="cohsp_1.html#E10:148_2_1_3"><i><font color="Green">E92</font></i></a></i>;<br/><a NAME="E12:148_2_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="cohsp_1.html#K14" target="_self">U+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="cohsp_1.html#K16" target="_self">"\/"</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="cohsp_1.html#T82" target="_self">Th82</a></i>;<br/><a NAME="E13:148_2_1_3"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>7</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="txt" href="cohsp_1.html#E3:148_2_1_3"><i><font color="Green">E89</font></i></a>, <a class="txt" href="cohsp_1.html#E8:148_2_1_3"><i><font color="Green">E91</font></i></a>, <a class="ref" href="cohsp_1.html#T87" target="_self">Th87</a></i>;<br/><b>hence </b><a NAME="E14:148_2_1_3"/>
<font color="Maroon">c<sub>7</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="coh_sp.html#T1">COH_SP:1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:148_2_1"/><b>then </b><i><font color="Green">E89</font></i>: 
 <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="cohsp_1.html#D1" target="_self">Def1</a></i>;<br/>
<div><i><font color="Green">E90</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="148_2_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:148_2_1_4"/>
 <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> =  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/><b>consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font>;<br/><b>consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:148_2_1_4"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></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">c<sub>10</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:148_2_1_4"/><i><font color="Green">E92</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Maroon">c<sub>10</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="cohsp_1.html#E5:148_2_1_4"><i><font color="Green">E91</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E9:148_2_1_4"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E7:148_2_1_4"><i><font color="Green">E92</font></i></a></i>;<br/><b>assume </b><a NAME="E10:148_2_1_4"/>
 <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> =  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/><b>consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>12</sub></font>;<br/><b>consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E14:148_2_1_4"/><i><font color="Green">E94</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></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">c<sub>15</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E16:148_2_1_4"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="cohsp_1.html#E14:148_2_1_4"><i><font color="Green">E94</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>16</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E18:148_2_1_4"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E16:148_2_1_4"><i><font color="Green">E95</font></i></a></i>;<br/><a NAME="E19:148_2_1_4"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="cohsp_1.html#K16" target="_self">"\/"</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E5:148_2_1"><i><font color="Green">E87</font></i></a>, <a class="txt" href="cohsp_1.html#E9:148_2_1_4"><i><font color="Green">E93</font></i></a>, <a class="txt" href="cohsp_1.html#E18:148_2_1_4"><i><font color="Green">E96</font></i></a></i>;<br/><a NAME="E20:148_2_1_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="cohsp_1.html#K14" target="_self">U+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="cohsp_1.html#K16" target="_self">"\/"</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="cohsp_1.html#T82" target="_self">Th82</a></i>;<br/><a NAME="E21:148_2_1_4"/><b>then </b>
( ( <font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>16</sub></font> ) or ( <font color="Maroon">c<sub>11</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>14</sub></font> ) )
 <b>by </b><i><a class="txt" href="cohsp_1.html#E5:148_2_1_4"><i><font color="Green">E91</font></i></a>, <a class="txt" href="cohsp_1.html#E7:148_2_1_4"><i><font color="Green">E92</font></i></a>, <a class="txt" href="cohsp_1.html#E14:148_2_1_4"><i><font color="Green">E94</font></i></a>, <a class="txt" href="cohsp_1.html#E16:148_2_1_4"><i><font color="Green">E95</font></i></a>, <a class="ref" href="cohsp_1.html#T87" target="_self">Th87</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E22:148_2_1_4"/>
contradiction
 ;<br/></div><b>end;</b></div>
<a NAME="E11:148_2_1"/>
<span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></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<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>4</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="148_2_1_5"><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">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:148_2_1_5_1"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">c<sub>7</sub></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">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></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<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span> </span>)</span>
 ;<br/><a NAME="E2:148_2_1_5_1"/><b>then </b><i><font color="Green">E92</font></i>: 
( <font color="Maroon">c<sub>7</sub></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">c<sub>7</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; ( ( <font color="Maroon">c<sub>7</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>7</sub></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<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span>  ) or ( <font color="Maroon">c<sub>7</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">c<sub>7</sub></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<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span>  ) ) )
 <b>by </b><i><a class="ref" href="cohsp_1.html#T76" target="_self">Th76</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="148_2_1_5_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_2_1_5_1_1"/>
( ( <font color="Maroon">c<sub>7</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>7</sub></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<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span>  ) or ( <font color="Maroon">c<sub>7</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">c<sub>7</sub></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<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span>  ) )
 </span><b>by </b><i><a class="txt" href="cohsp_1.html#E1:148_2_1_5_1"><i><font color="Green">E91</font></i></a>, <a class="ref" href="cohsp_1.html#T76" target="_self">Th76</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="148_2_1_5_1_1_1"><b>suppose </b></a><a NAME="E1:148_2_1_5_1_1_1"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>7</sub></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<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span>  )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:148_2_1_5_1_1_1"/><i><font color="Green">E94</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></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">c<sub>9</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:148_2_1_5_1_1_1"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Maroon">c<sub>9</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="cohsp_1.html#E3:148_2_1_5_1_1_1"><i><font color="Green">E94</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E7:148_2_1_5_1_1_1"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E5:148_2_1_5_1_1_1"><i><font color="Green">E95</font></i></a></i>;<br/><a NAME="E8:148_2_1_5_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E2:148_2_1_5_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="cohsp_1.html#E1:148_2_1_5_1_1_1"><i><font color="Green">E93</font></i></a>, <a class="txt" href="cohsp_1.html#E3:148_2_1_5_1_1_1"><i><font color="Green">E94</font></i></a>, <a class="txt" href="cohsp_1.html#E5:148_2_1_5_1_1_1"><i><font color="Green">E95</font></i></a>, <a class="ref" href="cohsp_1.html#T77" target="_self">Th77</a></i>;<br/><b>hence </b><a NAME="E9:148_2_1_5_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E7:148_2_1_5_1_1_1"><i><font color="Green">E96</font></i></a>, <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_2_1_5_1_1_2"><b>suppose </b></a><a NAME="E1:148_2_1_5_1_1_2"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">c<sub>7</sub></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<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span>  )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:148_2_1_5_1_1_2"/><i><font color="Green">E94</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></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">c<sub>9</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E5:148_2_1_5_1_1_2"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="cohsp_1.html#E3:148_2_1_5_1_1_2"><i><font color="Green">E94</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:148_2_1_5_1_1_2"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E5:148_2_1_5_1_1_2"><i><font color="Green">E95</font></i></a></i>;<br/><a NAME="E8:148_2_1_5_1_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E2:148_2_1_5_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="cohsp_1.html#E1:148_2_1_5_1_1_2"><i><font color="Green">E93</font></i></a>, <a class="txt" href="cohsp_1.html#E3:148_2_1_5_1_1_2"><i><font color="Green">E94</font></i></a>, <a class="txt" href="cohsp_1.html#E5:148_2_1_5_1_1_2"><i><font color="Green">E95</font></i></a>, <a class="ref" href="cohsp_1.html#T78" target="_self">Th78</a></i>;<br/><b>hence </b><a NAME="E9:148_2_1_5_1_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E7:148_2_1_5_1_1_2"><i><font color="Green">E96</font></i></a>, <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">c<sub>7</sub></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_2_1_5"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:148_2_1_5"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E5:148_2_1_5"/>
<font color="Maroon">c<sub>8</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="cohsp_1.html#K16" target="_self">"\/"</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="cohsp_1.html#E5:148_2_1"><i><font color="Green">E87</font></i></a>, <a class="txt" href="cohsp_1.html#E4:148_2_1_5"><i><font color="Green">E91</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>10</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E7:148_2_1_5"/><i><font color="Green">E92</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Maroon">c<sub>10</sub></font> &amp; ( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 <b>by </b><i><a class="ref" href="cohsp_1.html#T88" target="_self">Th88</a></i>;<br/>
<a NAME="E8:148_2_1_5"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span>  &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span>  )
 
<b>by </b><i><a class="txt" href="cohsp_1.html#E4:148_2_1_5"><i><font color="Green">E91</font></i></a>, <a class="txt" href="cohsp_1.html#E7:148_2_1_5"><i><font color="Green">E92</font></i></a></i>;<br/>
<a NAME="E9:148_2_1_5"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></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<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span>  &amp; <font color="Maroon">c<sub>10</sub></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<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></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_2_1_5"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Maroon">c<sub>10</sub></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">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></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<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span> </span>)</span>
 
<b>by </b><i><a class="ref" href="cohsp_1.html#T79" target="_self">Th79</a></i>;<br/>
<b>hence </b><a NAME="E11:148_2_1_5"/>
<font color="Maroon">c<sub>7</sub></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">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></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<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="cohsp_1.html#K14" target="_self">U+</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span> </span>)</span>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E4:148_2_1_5"><i><font color="Green">E91</font></i></a>, <a class="txt" href="cohsp_1.html#E7:148_2_1_5"><i><font color="Green">E92</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:148_2_1"/>
 <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="cohsp_1.html#K16" target="_self">"\/"</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E7:148_2_1"><i><font color="Green">E88</font></i></a>, <a class="txt" href="cohsp_1.html#E9:148_2_1"><i><font color="Green">E89</font></i></a>, <a class="txt" href="cohsp_1.html#E10:148_2_1"><i><font color="Green">E90</font></i></a>, <a class="ref" href="cohsp_1.html#T87" target="_self">Th87</a></i>;<br/>


</div>
