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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(),   <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>()] means ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">a<sub>1</sub></font>] &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] ) );<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <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>()] means for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">a<sub>1</sub></font>];<br/>
<a NAME="E1:37_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="setwiseo.html#K1">{}.</a> <font color="Maroon">F<sub>1</sub></font>()]
 
;<br/>
<div><i><font color="Green">E13</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1"><b>now </b></a><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/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>assume </b><a NAME="E1:37_1_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>1</sub></font>]
 ;<br/><b>thus </b><a NAME="E2:37_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>1</sub></font> <a href="setwiseo.html#K5">\/</a> <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="setwiseo.html#K2">.}</a></span>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<b>assume </b><a NAME="E1:37_1_1_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_1_1_1_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 </span><b>by </b><i><a class="txt" href="fraenkel.html#E1:37_1_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="setwiseo.html#T6">SETWISEO:6</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:37_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E3:37_1_1_1_1_1_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E4:37_1_1_1_1_1_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>]
 <b>and </b><br/><a NAME="E5:37_1_1_1_1_1_1"/><i><font color="Green">E18</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() 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="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font>] holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 <b>by </b><i><a class="txt" href="fraenkel.html#E1:37_1_1"><i><font color="Green">E14</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_1_1_1_1_1_1_1_1"/>
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>] or  not <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>] )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:37_1_1_1_1_1_1_1_1_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>]
 ;<br/><div class="add"><b>thus </b><a NAME="E2:37_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="setwiseo.html#T6">SETWISEO:6</a></i>;<br/><b>thus </b><a NAME="E3:37_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>]
 <b>by </b><i><a class="txt" href="fraenkel.html#E2:37"><i><font color="Green">E11</font></i></a>, <a class="txt" href="fraenkel.html#E4:37_1_1_1_1_1_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="fraenkel.html#E1:37_1_1_1_1_1_1_1_1_1"><i><font color="Green">E19</font></i></a></i>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>assume </b><b>that </b><br/><a NAME="E4:37_1_1_1_1_1_1_1_1_1"/><i><font color="Green">E20</font></i>: 
<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="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>and </b><br/><a NAME="E5:37_1_1_1_1_1_1_1_1_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>]
 ;<br/><a NAME="E6:37_1_1_1_1_1_1_1_1_1"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="fraenkel.html#E4:37_1_1_1_1_1_1_1_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="setwiseo.html#T6">SETWISEO:6</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:37_1_1_1_1_1_1_1_1_1_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E2:37_1_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>]
 <b>by </b><i><a class="txt" href="fraenkel.html#E2:37"><i><font color="Green">E11</font></i></a>, <a class="txt" href="fraenkel.html#E1:37_1_1_1_1_1_1_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="fraenkel.html#E5:37_1_1_1_1_1_1_1_1_1"><i><font color="Green">E21</font></i></a></i>;<br/><a NAME="E3:37_1_1_1_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>]
 <b>by </b><i><a class="txt" href="fraenkel.html#E5:37_1_1_1_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="fraenkel.html#E1:37_1_1_1_1_1_1_1_1_1_1"><i><font color="Green">E23</font></i></a></i>;<br/><b>hence </b><a NAME="E4:37_1_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>]
 <b>by </b><i><a class="txt" href="fraenkel.html#E2:37"><i><font color="Green">E11</font></i></a>, <a class="txt" href="fraenkel.html#E1:37_1_1_1_1_1_1_1_1_1"><i><font color="Green">E19</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E8:37_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>]
 <b>by </b><i><a class="txt" href="fraenkel.html#E1:37"><i><font color="Green">E10</font></i></a>, <a class="txt" href="fraenkel.html#E6:37_1_1_1_1_1_1_1_1_1"><i><font color="Green">E22</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:37_1_1_1_1_1_1_1_1_2"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>]
 ;<br/><div class="add"><b>thus </b><a NAME="E2:37_1_1_1_1_1_1_1_1_2"/>
<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="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="fraenkel.html#E3:37_1_1_1_1_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>thus </b><a NAME="E3:37_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>]
 <b>by </b><i><a class="txt" href="fraenkel.html#E4:37_1_1_1_1_1_1"><i><font color="Green">E17</font></i></a></i>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>assume </b><b>that </b><br/><a NAME="E4:37_1_1_1_1_1_1_1_1_2"/><i><font color="Green">E20</font></i>: 
<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="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>and </b><br/><a NAME="E5:37_1_1_1_1_1_1_1_1_2"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>]
 ;<br/><a NAME="E6:37_1_1_1_1_1_1_1_1_2"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="fraenkel.html#E4:37_1_1_1_1_1_1_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="setwiseo.html#T6">SETWISEO:6</a></i>;<br/><b>hence </b><a NAME="E7:37_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>]
 <b>by </b><i><a class="txt" href="fraenkel.html#E5:37_1_1_1_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="fraenkel.html#E1:37_1_1_1_1_1_1_1_1_2"><i><font color="Green">E19</font></i></a>, <a class="txt" href="fraenkel.html#E5:37_1_1_1_1_1_1_1_1_2"><i><font color="Green">E21</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:37_1_1_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="setwiseo.html#K5">\/</a> <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="setwiseo.html#K2">.}</a></span>]
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:37_1_1_1_1_1_2"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_1_1_1_1_1_2_1_1"/>
( ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <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="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>] ) or for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/> not <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>] )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:37_1_1_1_1_1_2_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <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="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>] )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E3:37_1_1_1_1_1_2_1_1_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E4:37_1_1_1_1_1_2_1_1_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>]
 <b>by </b><i><a class="txt" href="fraenkel.html#E1:37_1_1_1_1_1_2"><i><font color="Green">E16</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E6:37_1_1_1_1_1_2_1_1_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E7:37_1_1_1_1_1_2_1_1_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>]
 <b>and </b><br/><a NAME="E8:37_1_1_1_1_1_2_1_1_1"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() 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="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font>] holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 <b>by </b><i><a class="txt" href="fraenkel.html#E1:37_1_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="fraenkel.html#E3:37_1_1_1_1_1_2_1_1_1"><i><font color="Green">E17</font></i></a></i>;<br/><b>take </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font>;<br/><b>thus </b><a NAME="E9:37_1_1_1_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="fraenkel.html#E6:37_1_1_1_1_1_2_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>thus </b><a NAME="E10:37_1_1_1_1_1_2_1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>]
 <b>by </b><i><a class="txt" href="fraenkel.html#E2:37"><i><font color="Green">E11</font></i></a>, <a class="txt" href="fraenkel.html#E4:37_1_1_1_1_1_2_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="fraenkel.html#E7:37_1_1_1_1_1_2_1_1_1"><i><font color="Green">E20</font></i></a></i>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>assume </b><b>that </b><br/><a NAME="E11:37_1_1_1_1_1_2_1_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>and </b><br/><a NAME="E12:37_1_1_1_1_1_2_1_1_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font>]
 ;<br/><a NAME="E13:37_1_1_1_1_1_2_1_1_1"/>
( <font color="Maroon">c<sub>7</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="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="fraenkel.html#E11:37_1_1_1_1_1_2_1_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="setwiseo.html#T6">SETWISEO:6</a></i>;<br/><b>hence </b><a NAME="E14:37_1_1_1_1_1_2_1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>]
 <b>by </b><i><a class="txt" href="fraenkel.html#E2:37"><i><font color="Green">E11</font></i></a>, <a class="txt" href="fraenkel.html#E1:37_1_1_1_1_1_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="fraenkel.html#E4:37_1_1_1_1_1_2_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="fraenkel.html#E7:37_1_1_1_1_1_2_1_1_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="fraenkel.html#E8:37_1_1_1_1_1_2_1_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="fraenkel.html#E12:37_1_1_1_1_1_2_1_1_1"><i><font color="Green">E23</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:37_1_1_1_1_1_2_1_1_2"/><i><font color="Green">E17</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/> not <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>]
 ;<br/><div class="add"><b>thus </b><a NAME="E2:37_1_1_1_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="setwiseo.html#T6">SETWISEO:6</a></i>;<br/><b>thus </b><a NAME="E3:37_1_1_1_1_1_2_1_1_2"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font>]
 <b>by </b><i><a class="txt" href="fraenkel.html#E1:37"><i><font color="Green">E10</font></i></a></i>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>assume </b><b>that </b><br/><a NAME="E4:37_1_1_1_1_1_2_1_1_2"/><i><font color="Green">E19</font></i>: 
<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="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>and </b><br/><a NAME="E5:37_1_1_1_1_1_2_1_1_2"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>]
 ;<br/><a NAME="E6:37_1_1_1_1_1_2_1_1_2"/>
not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="fraenkel.html#E1:37_1_1_1_1_1_2_1_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="fraenkel.html#E5:37_1_1_1_1_1_2_1_1_2"><i><font color="Green">E20</font></i></a></i>;<br/><b>hence </b><a NAME="E7:37_1_1_1_1_1_2_1_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>]
 <b>by </b><i><a class="txt" href="fraenkel.html#E3:37_1_1_1_1_1_2_1_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="fraenkel.html#E4:37_1_1_1_1_1_2_1_1_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="setwiseo.html#T6">SETWISEO:6</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:37_1_1_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="setwiseo.html#K5">\/</a> <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="setwiseo.html#K2">.}</a></span>]
 <b>by </b><i><a class="txt" href="fraenkel.html#E1:37_1_1_1_1_1_2"><i><font color="Green">E16</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:37_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="setwiseo.html#K5">\/</a> <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="setwiseo.html#K2">.}</a></span>]
 ;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E3:37_1"/>
for <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>() holds <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="setwiseo.html#S4">SETWISEO:sch 4</a>(<a class="txt" href="fraenkel.html#E1:37_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="fraenkel.html#E2:37_1"><i><font color="Green">E13</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:37_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>() holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>] &amp; ( for <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>2</sub></font>] holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>] ) )
 ;<br/>


</div>
