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

<b>assume </b><a NAME="E1:42_1"/><i><font color="Green">E12</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()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>3</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> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font>] )
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>1</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>();<br/>
<b>consider </b><font color="Maroon">c<sub>2</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="E4:42_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>3</sub></font>()
 <b>and </b><br/><a NAME="E5:42_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>]
 <b>by </b><i><a class="txt" href="fraenkel.html#E1:42_1"><i><font color="Green">E12</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">{<span class="default"> <span class="p2">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] </span>}</span>  where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>3</sub></font>() </span>}</span> ;<br/>
<a NAME="E6:42_1"/>
<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">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span>  <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] </span>}</span>  where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>3</sub></font>() </span>}</span> 
 
<b>by </b><i><a class="txt" href="fraenkel.html#E4:42_1"><i><font color="Green">E13</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">{<span class="default"> <span class="p2">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] </span>}</span>  where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>3</sub></font>() </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:42_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</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">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E3:42_1_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</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">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>6</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 <b>and </b><br/><a NAME="E4:42_1_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>3</sub></font>()
 ;<br/><b>consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E6:42_1_1"/><i><font color="Green">E16</font></i>: 
<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#E1:42"><i><font color="Green">E11</font></i></a>, <a class="txt" href="fraenkel.html#E4:42_1_1"><i><font color="Green">E15</font></i></a></i>;<br/><a NAME="E7:42_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="fraenkel.html#E3:42_1_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="fraenkel.html#E6:42_1_1"><i><font color="Green">E16</font></i></a></i>;<br/><b>hence </b><a NAME="E8:42_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E10:42_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E11:42_1"/><i><font color="Green">E14</font></i>: 
for <font color="Olive">b<sub>1</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> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="wellord2.html#T28">WELLORD2:28</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>();<br/>
<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="hidden.html#M1">set</a> ] means ( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>3</sub></font>() implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</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">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span>  ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>3</sub></font>() or <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> ) );<br/>
<div><i><font color="Green">E15</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_2"><b>now </b></a><div class="add"><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/><a NAME="E1:42_1_2"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>3</sub></font>() implies <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>3</sub></font>() )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_2_1"><b>now </b></a><div class="add"><b>set </b><font color="Maroon">c<sub>8</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">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>7</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span> ;<br/><b>assume </b><a NAME="E1:42_1_2_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>3</sub></font>()
 ;<br/><a NAME="E2:42_1_2_1"/><b>then </b>
<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">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>7</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E3:42_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</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">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>7</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span>  <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">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>7</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 <b>by </b><i><a class="txt" href="fraenkel.html#E11:42_1"><i><font color="Green">E14</font></i></a></i>;<br/><a NAME="E4:42_1_2_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>7</sub></font>,<font color="Olive">b<sub>2</sub></font>] </span>}</span>  <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>7</sub></font>,<font color="Olive">b<sub>1</sub></font>] )
 ;<br/><b>hence </b><a NAME="E5:42_1_2_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</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">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>7</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span>  is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>()
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:42_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>7</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 <b>by </b><i><a class="txt" href="fraenkel.html#E1:42_1_2"><i><font color="Green">E16</font></i></a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E15:42_1"/><i><font color="Green">E16</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>() holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="txt" href="fraenkel.html#E13:42_1"><i><font color="Green">E15</font></i></a>);<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</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="E1:42_1_3"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>3</sub></font>()
 ;<br/><a NAME="E2:42_1_3"/><b>then </b><i><font color="Green">E18</font></i>: 
<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">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>8</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E3:42_1_3"/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</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">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>8</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 <b>by </b><i><a class="txt" href="fraenkel.html#E15:42_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="fraenkel.html#E1:42_1_3"><i><font color="Green">E17</font></i></a></i>;<br/><a NAME="E4:42_1_3"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <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">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>8</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 <b>by </b><i><a class="txt" href="fraenkel.html#E11:42_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="fraenkel.html#E2:42_1_3"><i><font color="Green">E18</font></i></a></i>;<br/><a NAME="E5:42_1_3"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>8</sub></font>,<font color="Olive">b<sub>1</sub></font>] )
 ;<br/><b>hence </b><a NAME="E6:42_1_3"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font>]
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E17:42_1"/>
contradiction
 <b>by </b><i><a class="txt" href="fraenkel.html#E1:42_1"><i><font color="Green">E12</font></i></a></i>;<br/>


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