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

<b>assume </b><a NAME="E1:42_1"/><i><font color="Green">E18</font></i>: 
for <font color="Olive">ff</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">t</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Olive">t</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">t</font>,<font color="Olive">ff</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t</font>] )
 ;<br/>

<b>consider </b><font color="Maroon">f</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">t</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">E19</font></i>: 
<font color="Maroon">t</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">t</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font>]
 <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">M</font> = <span class="p1">{<span class="default"> <span class="p2">{<span class="default"> <font color="Olive">ff</font> where <font color="Olive">ff</font> 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">tt</font>,<font color="Olive">ff</font>] </span>}</span>  where <font color="Olive">tt</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Olive">tt</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">ff</font> where <font color="Olive">ff</font> 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">t</font>,<font color="Olive">ff</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">ff</font> where <font color="Olive">ff</font> 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">tt</font>,<font color="Olive">ff</font>] </span>}</span>  where <font color="Olive">tt</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Olive">tt</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>3</sub></font>() </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">M</font> = <span class="p1">{<span class="default"> <span class="p2">{<span class="default"> <font color="Olive">ff</font> where <font color="Olive">ff</font> 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">tt</font>,<font color="Olive">ff</font>] </span>}</span>  where <font color="Olive">tt</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Olive">tt</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">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:42_1_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 ;<br/><b>then consider </b><font color="Maroon">t</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">E31</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">ff</font> where <font color="Olive">ff</font> 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">t</font>,<font color="Olive">ff</font>] </span>}</span> 
 <b>and </b><br/><a NAME="E4:42_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>3</sub></font>()
 ;<br/><b>consider </b><font color="Maroon">ff</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">E37</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">t</font>,<font color="Maroon">ff</font>]
 <b>by </b><i>, </i>;<br/><a NAME="E7:42_1_1"/>
<font color="Maroon">ff</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="txt" href="fraenkel.html#E1:42"><i><font color="Green">E16</font></i></a></i>;<br/><b>hence </b><a NAME="E8:42_1_1"/>
<font color="Maroon">X</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">Choice</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">Choice</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font>
 <b>and </b><br/><a NAME="E11:42_1"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> holds <br/><font color="Maroon">Choice</font> <a href="funct_1.html#K1">.</a> <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font>
 <b>by </b><i><a class="ref" href="wellord2.html#T28">WELLORD2:28</a></i>;<br/>
<b>consider </b><font color="Maroon">b</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> ] <b>means </b>( ( <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">Choice</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <font color="Olive">ff</font> where <font color="Olive">ff</font> 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">ff</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">b</font> ) );<br/>
<div><i><font color="Green">E81</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">t</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">E82</font></i>: 
( <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>3</sub></font>() implies <font color="Maroon">t</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">s</font> = <span class="p1">{<span class="default"> <font color="Olive">ff</font> where <font color="Olive">ff</font> 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">t</font>,<font color="Olive">ff</font>] </span>}</span> ;<br/><b>assume </b><a NAME="E1:42_1_2_1"/>
<font color="Maroon">t</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">ff</font> where <font color="Olive">ff</font> 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">t</font>,<font color="Olive">ff</font>] </span>}</span>  <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 ;<br/><a NAME="E3:42_1_2_1"/><b>then </b>
<font color="Maroon">Choice</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <font color="Olive">ff</font> where <font color="Olive">ff</font> 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">t</font>,<font color="Olive">ff</font>] </span>}</span>  <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">ff</font> where <font color="Olive">ff</font> 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">t</font>,<font color="Olive">ff</font>] </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E4:42_1_2_1"/><b>then </b>
 ex <font color="Olive">ff</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Maroon">Choice</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <font color="Olive">ff</font> where <font color="Olive">ff</font> 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">t</font>,<font color="Olive">ff</font>] </span>}</span>  <a href="hidden.html#R1">=</a> <font color="Olive">ff</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">t</font>,<font color="Olive">ff</font>] )
 ;<br/><b>hence </b><a NAME="E5:42_1_2_1"/>
<font color="Maroon">Choice</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <font color="Olive">ff</font> where <font color="Olive">ff</font> 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">t</font>,<font color="Olive">ff</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">y</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">t</font>,<font color="Olive">y</font>]
 <b>by </b><i><a class="txt" href="fraenkel.html#E1:42"><i><font color="Green">E16</font></i></a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</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">E84</font></i>: 
for <font color="Olive">x</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">x</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</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">t</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">E85</font></i>: 
<font color="Maroon">t</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">E86</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">ff</font> where <font color="Olive">ff</font> 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">t</font>,<font color="Olive">ff</font>] </span>}</span>  <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 ;<br/><a NAME="E3:42_1_3"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Choice</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <font color="Olive">ff</font> where <font color="Olive">ff</font> 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">t</font>,<font color="Olive">ff</font>] </span>}</span> 
 <b>by </b><i><a class="txt" href="fraenkel.html#E1:42"><i><font color="Green">E16</font></i></a>, </i>;<br/><a NAME="E4:42_1_3"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">ff</font> where <font color="Olive">ff</font> 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">t</font>,<font color="Olive">ff</font>] </span>}</span> 
 <b>by </b><i>, <a class="txt" href="fraenkel.html#E1:42_1"><i><font color="Green">E18</font></i></a></i>;<br/><a NAME="E5:42_1_3"/><b>then </b>
 ex <font color="Olive">ff</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Olive">ff</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">t</font>,<font color="Olive">ff</font>] )
 ;<br/><b>hence </b><a NAME="E6:42_1_3"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">t</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font>]
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E17:42_1"/>
contradiction
 <b>by </b><i/>;<br/>


</div>
