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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">F<sub>1</sub></font>();<br/>

<a NAME="E1:27_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a>
 
;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  implies ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) );<br/>
<a NAME="E2:27_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1">{}</a> ]
 
;<br/>
<a NAME="E3:27_1"/><i><font color="Green">E21</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>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:27_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E2:27_1_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E3:27_1_1"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  implies ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) )
 <b>and </b><br/><a NAME="E4:27_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="txt" href="setwiseo.html#E2:27_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="setwiseo.html#T16" target="_self">Th16</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>2</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="txt" href="setwiseo.html#E1:27_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="setwiseo.html#T14" target="_self">Th14</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font> <a href="setwiseo.html#K5" target="_self">\/</a> <span class="p1"><a href="setwiseo.html#K2" target="_self">{.</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="setwiseo.html#K2" target="_self">.}</a></span>
;<br/>

<b>thus </b><a NAME="E7:27_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="setwiseo.html#K5" target="_self">\/</a> <span class="p1"><a href="setwiseo.html#K2" target="_self">{.</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="setwiseo.html#K2" target="_self">.}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E8:27_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<span class="p1"><a href="setwiseo.html#K2" target="_self">{.</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="setwiseo.html#K2" target="_self">.}</a></span>]
 
<b>by </b><i><a class="txt" href="setwiseo.html#E1:27"><i><font color="Green">E17</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_1_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_1"><b>suppose </b></a><a NAME="E1:27_1_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:27_1_1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font> <a href="setwiseo.html#K5" target="_self">\/</a> <span class="p1"><a href="setwiseo.html#K2" target="_self">{.</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="setwiseo.html#K2" target="_self">.}</a></span>]
 <b>by </b><i><a class="txt" href="setwiseo.html#E1:27"><i><font color="Green">E17</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_2"><b>suppose </b></a><a NAME="E1:27_1_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:27_1_1_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font> <a href="setwiseo.html#K5" target="_self">\/</a> <span class="p1"><a href="setwiseo.html#K2" target="_self">{.</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="setwiseo.html#K2" target="_self">.}</a></span>]
 <b>by </b><i><a class="txt" href="setwiseo.html#E2:27"><i><font color="Green">E18</font></i></a>, <a class="txt" href="setwiseo.html#E3:27_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="setwiseo.html#E8:27_1_1"><i><font color="Green">E25</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E4:27_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="finset_1.html#S2">FINSET_1:sch 2</a>(<a class="txt" href="setwiseo.html#E1:27_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="setwiseo.html#E2:27_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="setwiseo.html#E3:27_1"><i><font color="Green">E21</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:27_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font>] )
 ;<br/>


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