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

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_1_1"/>
( <font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or ( <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">F<sub>2</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="12_1_1_1"><b>suppose </b></a><a NAME="E1:12_1_1_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><div><i><font color="Green">E23</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_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="E1:12_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/><a NAME="E2:12_1_1_1_1"/><b>then </b>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</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="Maroon">x</font>,<font color="Olive">y</font>] )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:12_1_1_1_1"/>
contradiction
 <b>by </b><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>();<br/><b>take </b>
<font color="Maroon">f</font>
;<br/><b>thus </b><a NAME="E4:12_1_1_1"/>
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">F<sub>1</sub></font>() holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_2"><b>suppose </b></a><a NAME="E1:12_1_1_2"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><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>take </b>
<font color="Maroon">f</font>
;<br/><b>thus </b><a NAME="E3:12_1_1_2"/>
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">F<sub>1</sub></font>() holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_3"><b>suppose </b></a><a NAME="E1:12_1_1_3"/>
( <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">X</font> = <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">Y</font> = <font color="Maroon">F<sub>2</sub></font>() as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<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">Y</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">X</font> : verum </span>}</span> ;<br/><b>consider </b><font color="Maroon">x0</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/><a NAME="E4:12_1_1_3"/>
<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">Y</font> : <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">x0</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">Y</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">X</font> : verum </span>}</span> 
 ;<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">Y</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">X</font> : verum </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="12_1_1_3_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="E1:12_1_1_3_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">X</font><b> such that </b><br/><a NAME="E3:12_1_1_3_1"/><i><font color="Green">E34</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">Y</font> : <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">t</font>,<font color="Olive">ff</font>] </span>}</span> 
 ;<br/><b>consider </b><font color="Maroon">ff</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:12_1_1_3_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">ff</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E6:12_1_1_3_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">t</font>,<font color="Maroon">ff</font>]
 <b>by </b><i/>;<br/><a NAME="E7:12_1_1_3_1"/>
<font color="Maroon">ff</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E8:12_1_1_3_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">C</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E8:12_1_1_3"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font>
 <b>and </b><br/><a NAME="E9:12_1_1_3"/><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">C</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>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">C</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">Y</font> : <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font>,<font color="Olive">ff</font>] </span>}</span> ;<br/><b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E11:12_1_1_3"/><i><font color="Green">E40</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E12:12_1_1_3"/><i><font color="Green">E41</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">X</font> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>)
 <b>from </b><i><a class="ref" href="funct_1.html#S3">FUNCT_1:sch 3</a>();<br/></i><a NAME="E13:12_1_1_3"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:12_1_1_3_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:12_1_1_3_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E4:12_1_1_3_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>set </b><font color="Maroon">y0</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">Y</font> : <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">y</font>,<font color="Olive">ff</font>] </span>}</span> ;<br/>
<a NAME="E5:12_1_1_3_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</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">Y</font> : <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">y</font>,<font color="Olive">ff</font>] </span>}</span> 
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E6:12_1_1_3_2"/>
<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">Y</font> : <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">y</font>,<font color="Olive">ff</font>] </span>}</span>  <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 
<b>by </b><i/>;<br/>
<a NAME="E7:12_1_1_3_2"/><b>then </b>
<font color="Maroon">x</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">Y</font> : <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">y</font>,<font color="Olive">ff</font>] </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:12_1_1_3_2"/><b>then </b>
 ex <font color="Olive">ff</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">ff</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">y</font>,<font color="Olive">ff</font>] )
 
;<br/>
<b>hence </b><a NAME="E9:12_1_1_3_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <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>by </b><i>, , <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/><b>take </b>
<font color="Maroon">f</font>
;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>set </b><font color="Maroon">x0</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">Y</font> : <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Olive">ff</font>] </span>}</span> ;<br/><b>assume </b><a NAME="E15:12_1_1_3"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/><a NAME="E16:12_1_1_3"/><b>then </b><i><font color="Green">E47</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">Y</font> : <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Olive">ff</font>] </span>}</span>  <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 ;<br/><a NAME="E17:12_1_1_3"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</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">Y</font> : <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Olive">ff</font>] </span>}</span> 
 <b>by </b><i>, </i>;<br/><a NAME="E18:12_1_1_3"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</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">Y</font> : <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Olive">ff</font>] </span>}</span> 
 <b>by </b><i>, </i>;<br/><a NAME="E19:12_1_1_3"/><b>then </b>
 ex <font color="Olive">ff</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> st <br/>( <font color="Olive">ff</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Olive">ff</font>] )
 ;<br/><b>hence </b><a NAME="E20:12_1_1_3"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>]
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
