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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">G</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font> st <br/>( ( for <font color="Olive">e</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>  st <font color="Olive">e</font> <a href="binop_1.html#R3">is_a_unity_wrt</a> <font color="Maroon">F</font> holds <br/><font color="Olive">G</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Olive">e</font> ) &amp; ( for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> holds  <font color="Olive">G</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> ) &amp; ( for <font color="Olive">B'</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font>  st <font color="Olive">B'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">B'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/>for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <font color="Olive">B'</font> holds <br/><font color="Olive">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">B'</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">G</font> <a href="funct_2.html#K3">.</a> <font color="Olive">B'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> ) );<br/>
<a NAME="E1:29_1_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="setwiseo.html#K1" target="_self">{}.</a> <font color="Maroon">X</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">e</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E2:29_1_1_1"/><i><font color="Green">E22</font></i>: 
(  ex <font color="Olive">e</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> st <font color="Olive">e</font> <a href="binop_1.html#R3">is_a_unity_wrt</a> <font color="Maroon">F</font> implies <font color="Maroon">e</font> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">F</font> )
 ;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>( ( for <font color="Olive">x'</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x'</font></span><a href="tarski.html#K1">}</a></span> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x'</font> ) &amp; ( ( for <font color="Olive">x'</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  holds not <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x'</font></span><a href="tarski.html#K1">}</a></span> ) implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> ) );<br/>
<div><i><font color="Green">E23</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font>;<br/><b>thus </b><a NAME="E1:29_1_1_1_1"/>
 ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">x</font>,<font color="Olive">y</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1_1_1"><b>proof </b></a><div class="add">

<div><i><font color="Green">E24</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">x'</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font><b> such that </b><a NAME="E1:29_1_1_1_1_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x'</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><b>take </b><font color="Maroon">y</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font>;<br/><b>thus </b><a NAME="E2:29_1_1_1_1_1_1"/>
for <font color="Olive">x'</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x'</font></span><a href="tarski.html#K1">}</a></span> holds <br/><font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x'</font>
 <b>by </b><i><a class="txt" href="setwiseo.html#E1:29_1_1_1_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="zfmisc_1.html#T6">ZFMISC_1:6</a></i>;<br/><b>thus </b><a NAME="E3:29_1_1_1_1_1_1"/>
( ( for <font color="Olive">x'</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  holds not <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x'</font></span><a href="tarski.html#K1">}</a></span> ) implies <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> )
 <b>by </b><i><a class="txt" href="setwiseo.html#E1:29_1_1_1_1_1"><i><font color="Green">E24</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:29_1_1_1_1_1_2"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">x'</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  holds not <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x'</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><b>take </b><font color="Maroon">y</font> = <font color="Maroon">e</font>;<br/><b>thus </b><a NAME="E2:29_1_1_1_1_1_2"/>
( ( for <font color="Olive">x'</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x'</font></span><a href="tarski.html#K1">}</a></span> holds <br/><font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x'</font> ) &amp; ( ( for <font color="Olive">x'</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  holds not <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x'</font></span><a href="tarski.html#K1">}</a></span> ) implies <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> ) )
 <b>by </b><i><a class="txt" href="setwiseo.html#E1:29_1_1_1_1_1"><i><font color="Green">E24</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:29_1_1_1_1_1"/>
 ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">x</font>,<font color="Olive">y</font>]
 <b>by </b><i><a class="txt" href="setwiseo.html#E3:29_1_1_1"><i><font color="Green">E23</font></i></a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>consider </b><font color="Maroon">G'</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font><b> such that </b><br/><a NAME="E5:29_1_1_1"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">B'</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">B'</font>,<font color="Maroon">G'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">B'</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="txt" href="setwiseo.html#E2:29_1_1_1"><i><font color="Green">E22</font></i></a>);<br/></i>
<b>take </b><font color="Maroon">G</font> = <font color="Maroon">G'</font>;<br/>

<b>thus </b><a NAME="E6:29_1_1_1"/>
for <font color="Olive">e</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>  st <font color="Olive">e</font> <a href="binop_1.html#R3">is_a_unity_wrt</a> <font color="Maroon">F</font> holds <br/><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Olive">e</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e'</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>;<br/>

<b>assume </b><a NAME="E1:29_1_1_1_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">e'</font> <a href="binop_1.html#R3">is_a_unity_wrt</a> <font color="Maroon">F</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">E</font> =  <a href="xboole_0.html#K1">{}</a>  as    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="finsub_1.html#T18">FINSUB_1:18</a></i>;<br/>
<a NAME="E3:29_1_1_1_2"/>
for <font color="Olive">x'</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  holds not <font color="Maroon">E</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x'</font></span><a href="tarski.html#K1">}</a></span>
 
;<br/>
<b>hence </b><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  = 
<font color="Maroon">e</font>
<b>by </b><i><a class="txt" href="setwiseo.html#E3:29_1_1_1"><i><font color="Green">E23</font></i></a></i>
<br/>.= 
<font color="Maroon">e'</font>
<b>by </b><i><a class="txt" href="setwiseo.html#E1:29_1_1"><i><font color="Green">E21</font></i></a>, , <a class="ref" href="binop_1.html#D8">BINOP_1:def 8</a></i>
;<br/><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E7:29_1_1_1"/>
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> holds  <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="setwiseo.html#K2" target="_self">{.</a><span class="default"><font color="Olive">x</font></span><a href="setwiseo.html#K2" target="_self">.}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>
 <b>by </b><i><a class="txt" href="setwiseo.html#E3:29_1_1_1"><i><font color="Green">E23</font></i></a></i>;<br/>

<b>thus </b><a NAME="E8:29_1_1_1"/>
for <font color="Olive">B'</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font>  st <font color="Olive">B'</font> <a href="tarski.html#R1">c=</a>  <a href="setwiseo.html#K1" target="_self">{}.</a> <font color="Maroon">X</font> &amp; <font color="Olive">B'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/>for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K1" target="_self">{}.</a> <font color="Maroon">X</font></span>)</span> <a href="setwiseo.html#K6" target="_self">\</a> <font color="Olive">B'</font> holds <br/><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">B'</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Olive">B'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:29_1_1"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">B'</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font><br/> for <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B'</font>] &amp; not <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Olive">B'</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B'</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">b</font></span><a href="tarski.html#K1">}</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">B</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/>



<b>given </b><font color="Maroon">G</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font><b> such that </b><a NAME="E1:29_1_1_2"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">e</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>  st <font color="Olive">e</font> <a href="binop_1.html#R3">is_a_unity_wrt</a> <font color="Maroon">F</font> holds <br/><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Olive">e</font>
 <b>and </b><br/><a NAME="E2:29_1_1_2"/><i><font color="Green">E63</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> holds  <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>
 <b>and </b><br/><a NAME="E3:29_1_1_2"/><i><font color="Green">E64</font></i>: 
for <font color="Olive">B'</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font>  st <font color="Olive">B'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> &amp; <font color="Olive">B'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/>for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="setwiseo.html#K6" target="_self">\</a> <font color="Olive">B'</font> holds <br/><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">B'</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Olive">B'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span>
 ;<br/>

<b>assume </b><a NAME="E4:29_1_1_2"/><i><font color="Green">E65</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>

<b>thus </b><a NAME="E5:29_1_1_2"/>
 ex <font color="Olive">G</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font> st <br/>( ( for <font color="Olive">e</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>  st <font color="Olive">e</font> <a href="binop_1.html#R3">is_a_unity_wrt</a> <font color="Maroon">F</font> holds <br/><font color="Olive">G</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Olive">e</font> ) &amp; ( for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> holds  <font color="Olive">G</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> ) &amp; ( for <font color="Olive">B'</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font>  st <font color="Olive">B'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Olive">B'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/>for <font color="Olive">x'</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">x'</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Olive">B'</font> holds <br/><font color="Olive">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">B'</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x'</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">G</font> <a href="funct_2.html#K3">.</a> <font color="Olive">B'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x'</font></span>)</span> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_1"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>( ( for <font color="Olive">C</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font>  st <font color="Olive">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">C</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">x</font></span><a href="setwiseo.html#K2" target="_self">.}</a></span> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Olive">C</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> ) &amp; ( ( for <font color="Olive">C</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font>  holds <br/>(  not <font color="Olive">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or  not <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> or  not <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">C</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">x</font></span><a href="setwiseo.html#K2" target="_self">.}</a></span> ) ) implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> ) );<br/>
<div><i><font color="Green">E67</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">B'</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font>;<br/><b>thus </b><a NAME="E1:29_1_1_2_1_1"/>
 ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">B'</font>,<font color="Olive">y</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_1_1_1"><b>proof </b></a><div class="add">

<div><i><font color="Green">E68</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">C</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font><b> such that </b><a NAME="E1:29_1_1_2_1_1_1_1"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> &amp; <font color="Maroon">B'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><b>take </b><font color="Maroon">y</font> = <font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>;<br/><a NAME="E2:29_1_1_2_1_1_1_1"/>
for <font color="Olive">C</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font>  st <font color="Olive">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> &amp; <font color="Maroon">B'</font> <a href="hidden.html#R1">=</a> <font color="Olive">C</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> holds <br/><font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Olive">C</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">C1</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E1:29_1_1_2_1_1_1_1_1"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">C1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">C1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> &amp; <font color="Maroon">B'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C1</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_1_1_1_1_1_1"><b>now </b></a><div class="add"><a NAME="E1:29_1_1_2_1_1_1_1_1_1"/><i><font color="Green">E72</font></i>: 
( not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font> )
 <b>by </b><i>, <a class="ref" href="setwiseo.html#T14" target="_self">Th14</a>, </i>;<br/><a NAME="E2:29_1_1_2_1_1_1_1_1_1"/>
( <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B'</font> &amp; <font color="Maroon">C1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B'</font> )
 <b>by </b><i><a class="ref" href="setwiseo.html#T14" target="_self">Th14</a>, , <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E3:29_1_1_2_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E73</font></i>: 
( <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B'</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">C1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B'</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T40">ZFMISC_1:40</a></i>;<br/><a NAME="E4:29_1_1_2_1_1_1_1_1_1"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">B'</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">B'</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">C1</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="setwiseo.html#T14" target="_self">Th14</a>, , <a class="ref" href="xboole_1.html#T40">XBOOLE_1:40</a></i>;<br/><a NAME="E5:29_1_1_2_1_1_1_1_1_1"/>
( <font color="Maroon">B'</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> &amp; <font color="Maroon">B'</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">C1</font> )
 <b>by </b><i><a class="ref" href="setwiseo.html#T16" target="_self">Th16</a>, <a class="txt" href="setwiseo.html#E17"><i><font color="Green">Lemma32</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><b>hence </b><a NAME="E6:29_1_1_2_1_1_1_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">C1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:29_1_1_2_1_1_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">C1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:29_1_1_2_1_1_1_1"/>
 ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">B'</font>,<font color="Olive">y</font>]
 <b>by </b><i><a class="ref" href="setwiseo.html#T14" target="_self">Th14</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_1_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:29_1_1_2_1_1_1_2"/><i><font color="Green">E75</font></i>: 
for <font color="Olive">C</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font>  holds <br/>(  not <font color="Olive">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or  not <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> or  not <font color="Maroon">B'</font> <a href="hidden.html#R1">=</a> <font color="Olive">C</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><b>take </b><font color="Maroon">y</font> = <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B'</font>;<br/><b>thus </b><a NAME="E2:29_1_1_2_1_1_1_2"/>
for <font color="Olive">C</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font>  st <font color="Olive">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> &amp; <font color="Maroon">B'</font> <a href="hidden.html#R1">=</a> <font color="Olive">C</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> holds <br/><font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Olive">C</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="setwiseo.html#T14" target="_self">Th14</a></i>;<br/><b>thus </b><a NAME="E3:29_1_1_2_1_1_1_2"/>
( ( for <font color="Olive">C</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font>  holds <br/>(  not <font color="Olive">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or  not <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> or  not <font color="Maroon">B'</font> <a href="hidden.html#R1">=</a> <font color="Olive">C</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> ) ) implies <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B'</font> )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:29_1_1_2_1_1_1"/>
 ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">B'</font>,<font color="Olive">y</font>]
 <b>by </b><i><a class="ref" href="setwiseo.html#T13" target="_self">Th13</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>consider </b><font color="Maroon">H</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font><b> such that </b><br/><a NAME="E3:29_1_1_2_1"/><i><font color="Green">E77</font></i>: 
for <font color="Olive">B'</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">B'</font>,<font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Olive">B'</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="ref" href="setwiseo.html#T12" target="_self">Th12</a>);<br/></i>
<b>take </b><font color="Maroon">J</font> = <font color="Maroon">H</font>;<br/>

<b>thus </b><a NAME="E4:29_1_1_2_1"/>
for <font color="Olive">e</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>  st <font color="Olive">e</font> <a href="binop_1.html#R3">is_a_unity_wrt</a> <font color="Maroon">F</font> holds <br/><font color="Maroon">J</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Olive">e</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_1_2"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">E</font> =  <a href="xboole_0.html#K1">{}</a>  as    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="finsub_1.html#T18">FINSUB_1:18</a></i>;<br/>
<a NAME="E2:29_1_1_2_1_2"/>
for <font color="Olive">C</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font>  holds <br/>(  not <font color="Olive">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or  not <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> or  not <font color="Maroon">E</font> <a href="hidden.html#R1">=</a> <font color="Olive">C</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 
;<br/>
<a NAME="E3:29_1_1_2_1_2"/><b>then </b>
<font color="Maroon">J</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">E</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">E</font>
 
<b>by </b><i><a class="ref" href="setwiseo.html#T13" target="_self">Th13</a></i>;<br/>
<b>hence </b><a NAME="E4:29_1_1_2_1_2"/>
for <font color="Olive">e</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>  st <font color="Olive">e</font> <a href="binop_1.html#R3">is_a_unity_wrt</a> <font color="Maroon">F</font> holds <br/><font color="Maroon">J</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Olive">e</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E5:29_1_1_2_1"/>
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> holds  <font color="Maroon">J</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x'</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_1_3_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">C</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font><b> such that </b><a NAME="E1:29_1_1_2_1_3_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E2:29_1_1_2_1_3_1"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E3:29_1_1_2_1_3_1"/><i><font color="Green">E81</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x'</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E4:29_1_1_2_1_3_1"/>
( <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x'</font></span><a href="tarski.html#K1">}</a></span> &amp; <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x'</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E5:29_1_1_2_1_3_1"/><b>then </b>
( <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x'</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> )
 <b>by </b><i><a class="ref" href="setwiseo.html#T14" target="_self">Th14</a>, <a class="ref" href="zfmisc_1.html#T24">ZFMISC_1:24</a>, <a class="ref" href="zfmisc_1.html#T39">ZFMISC_1:39</a></i>;<br/><a NAME="E6:29_1_1_2_1_3_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E7:29_1_1_2_1_3_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><font color="Maroon">J</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x'</font></span><a href="tarski.html#K1">}</a></span> = 
<font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="setwiseo.html#K2" target="_self">{.</a><span class="default"><font color="Maroon">x'</font></span><a href="setwiseo.html#K2" target="_self">.}</a></span>
<b>by </b><i><a class="ref" href="setwiseo.html#T13" target="_self">Th13</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font>
<b>by </b><i/>
;<br/><br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">B'</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E6:29_1_1_2_1"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">B'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>and </b><br/><a NAME="E7:29_1_1_2_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">B'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_1_4"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">C</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font><b> such that </b><a NAME="E1:29_1_1_2_1_4"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E2:29_1_1_2_1_4"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E3:29_1_1_2_1_4"/><i><font color="Green">E86</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E4:29_1_1_2_1_4"/>
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="setwiseo.html#E17"><i><font color="Green">Lemma32</font></i></a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E5:29_1_1_2_1_4"/><b>then </b>
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T39">ZFMISC_1:39</a></i>;<br/><a NAME="E6:29_1_1_2_1_4"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E7:29_1_1_2_1_4"/>
contradiction
 <b>by </b><i>, <a class="ref" href="setwiseo.html#T16" target="_self">Th16</a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:29_1_1_2_1"/><b>then </b><i><font color="Green">E87</font></i>: 
<font color="Maroon">J</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="setwiseo.html#K2" target="_self">{.</a><span class="default"><font color="Maroon">x</font></span><a href="setwiseo.html#K2" target="_self">.}</a></span>
 
<b>by </b><i><a class="ref" href="setwiseo.html#T13" target="_self">Th13</a></i>;<br/>
<b>let </b><font color="Maroon">x'</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E10:29_1_1_2_1"/>
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">B'</font>
 ;<br/>

<a NAME="E11:29_1_1_2_1"/><b>then </b><i><font color="Green">E88</font></i>: 
( <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> &amp; not <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_2_1_5"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1_1_2_1_5"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font> or  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_2_1_5_1"><b>suppose </b></a><a NAME="E1:29_1_1_2_1_5_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font>
 ;<br/><div class="add"><a NAME="E2:29_1_1_2_1_5_1"/><b>then </b><i><font color="Green">E90</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="setwiseo.html#T16" target="_self">Th16</a>, </i>;<br/><a NAME="E3:29_1_1_2_1_5_1"/><b>then </b><i><font color="Green">E91</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x'</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_2_1_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1_1_2_1_5_1_1"/>
( <font color="Maroon">B'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">B'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_2_1_5_1_1_1"><b>suppose </b></a><a NAME="E1:29_1_1_2_1_5_1_1_1"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">B'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><b>hence </b><font color="Maroon">J</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">B'</font> <a href="setwiseo.html#K5" target="_self">\/</a> <span class="p2"><a href="setwiseo.html#K2" target="_self">{.</a><span class="default"><font color="Maroon">x'</font></span><a href="setwiseo.html#K2" target="_self">.}</a></span></span>)</span> = 
<font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p2"><a href="setwiseo.html#K2" target="_self">{.</a><span class="default"><font color="Maroon">x'</font></span><a href="setwiseo.html#K2" target="_self">.}</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="setwiseo.html#T13" target="_self">Th13</a>, <a class="ref" href="setwiseo.html#T19" target="_self">Th19</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font></span>)</span>
<b>by </b><i><a class="ref" href="setwiseo.html#T8" target="_self">Th8</a>, <a class="ref" href="binop_1.html#D2">BINOP_1:def 2</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font></span>)</span>
<b>by </b><i>, , </i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_2_1_5_1_1_2"><b>suppose </b></a><a NAME="E1:29_1_1_2_1_5_1_1_2"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">B'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><b>set </b><font color="Maroon">C</font> = <span class="p1">(<span class="default"><font color="Maroon">B'</font> <a href="setwiseo.html#K6" target="_self">\</a> <span class="p2"><a href="setwiseo.html#K2" target="_self">{.</a><span class="default"><font color="Maroon">x</font></span><a href="setwiseo.html#K2" target="_self">.}</a></span></span>)</span> <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">x'</font></span><a href="setwiseo.html#K2" target="_self">.}</a></span>;<br/><a NAME="E2:29_1_1_2_1_5_1_1_2"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">B'</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="setwiseo.html#T14" target="_self">Th14</a>, <a class="ref" href="xboole_1.html#T43">XBOOLE_1:43</a></i>;<br/><a NAME="E3:29_1_1_2_1_5_1_1_2"/><b>then </b><i><font color="Green">E95</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">B'</font> <a href="setwiseo.html#K6" target="_self">\</a> <span class="p2"><a href="setwiseo.html#K2" target="_self">{.</a><span class="default"><font color="Maroon">x</font></span><a href="setwiseo.html#K2" target="_self">.}</a></span></span>)</span> <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">x'</font></span><a href="setwiseo.html#K2" target="_self">.}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="setwiseo.html#T18" target="_self">Th18</a>, </i>;<br/><a NAME="E4:29_1_1_2_1_5_1_1_2"/>
not <font color="Maroon">B'</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T39">ZFMISC_1:39</a></i>;<br/><a NAME="E5:29_1_1_2_1_5_1_1_2"/><b>then </b><i><font color="Green">E96</font></i>: 
<font color="Maroon">B'</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</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> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/><a NAME="E6:29_1_1_2_1_5_1_1_2"/>
not <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="setwiseo.html#T16" target="_self">Th16</a></i>;<br/><a NAME="E7:29_1_1_2_1_5_1_1_2"/><b>then </b><i><font color="Green">E97</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">B'</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="setwiseo.html#T18" target="_self">Th18</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><font color="Maroon">B'</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">x'</font></span><a href="setwiseo.html#K2" target="_self">.}</a></span> = 
<span class="p1">(<span class="default"><font color="Maroon">B'</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x'</font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="txt" href="setwiseo.html#E17"><i><font color="Green">Lemma32</font></i></a>, <a class="ref" href="zfmisc_1.html#T46">ZFMISC_1:46</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B'</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x'</font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="xboole_1.html#T39">XBOOLE_1:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B'</font> <a href="setwiseo.html#K6" target="_self">\</a> <span class="p3"><a href="setwiseo.html#K2" target="_self">{.</a><span class="default"><font color="Maroon">x</font></span><a href="setwiseo.html#K2" target="_self">.}</a></span></span>)</span> <a href="setwiseo.html#K5" target="_self">\/</a> <span class="p2"><a href="setwiseo.html#K2" target="_self">{.</a><span class="default"><font color="Maroon">x'</font></span><a href="setwiseo.html#K2" target="_self">.}</a></span></span>)</span> <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">x</font></span><a href="setwiseo.html#K2" target="_self">.}</a></span>
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>
;<br/><b>hence </b><font color="Maroon">J</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">B'</font> <a href="setwiseo.html#K5" target="_self">\/</a> <span class="p2"><a href="setwiseo.html#K2" target="_self">{.</a><span class="default"><font color="Maroon">x'</font></span><a href="setwiseo.html#K2" target="_self">.}</a></span></span>)</span> = 
<font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B'</font> <a href="setwiseo.html#K6" target="_self">\</a> <span class="p4"><a href="setwiseo.html#K2" target="_self">{.</a><span class="default"><font color="Maroon">x</font></span><a href="setwiseo.html#K2" target="_self">.}</a></span></span>)</span> <a href="setwiseo.html#K5" target="_self">\/</a> <span class="p3"><a href="setwiseo.html#K2" target="_self">{.</a><span class="default"><font color="Maroon">x'</font></span><a href="setwiseo.html#K2" target="_self">.}</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="setwiseo.html#T13" target="_self">Th13</a>, </i>
<br/>.= 
<font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">B'</font> <a href="setwiseo.html#K6" target="_self">\</a> <span class="p4"><a href="setwiseo.html#K2" target="_self">{.</a><span class="default"><font color="Maroon">x</font></span><a href="setwiseo.html#K2" target="_self">.}</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, , , </i>
<br/>.= 
<font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">B'</font> <a href="setwiseo.html#K6" target="_self">\</a> <span class="p3"><a href="setwiseo.html#K2" target="_self">{.</a><span class="default"><font color="Maroon">x</font></span><a href="setwiseo.html#K2" target="_self">.}</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="setwiseo.html#T11" target="_self">Th11</a>, <a class="ref" href="binop_1.html#D3">BINOP_1:def 3</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">B'</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="setwiseo.html#T8" target="_self">Th8</a>, <a class="ref" href="binop_1.html#D2">BINOP_1:def 2</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">B'</font> <a href="setwiseo.html#K6" target="_self">\</a> <span class="p4"><a href="setwiseo.html#K2" target="_self">{.</a><span class="default"><font color="Maroon">x</font></span><a href="setwiseo.html#K2" target="_self">.}</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font></span>)</span>
<b>by </b><i><a class="ref" href="setwiseo.html#T11" target="_self">Th11</a>, <a class="ref" href="binop_1.html#D3">BINOP_1:def 3</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B'</font> <a href="setwiseo.html#K6" target="_self">\</a> <span class="p4"><a href="setwiseo.html#K2" target="_self">{.</a><span class="default"><font color="Maroon">x</font></span><a href="setwiseo.html#K2" target="_self">.}</a></span></span>)</span> <a href="setwiseo.html#K5" target="_self">\/</a> <span class="p3"><a href="setwiseo.html#K2" target="_self">{.</a><span class="default"><font color="Maroon">x</font></span><a href="setwiseo.html#K2" target="_self">.}</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font></span>)</span>
<b>by </b><i><a class="ref" href="setwiseo.html#T13" target="_self">Th13</a>, , </i>
<br/>.= 
<font color="Maroon">F</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">B'</font> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T39">XBOOLE_1:39</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font></span>)</span>
<b>by </b><i><a class="txt" href="setwiseo.html#E17"><i><font color="Green">Lemma32</font></i></a>, <a class="ref" href="zfmisc_1.html#T46">ZFMISC_1:46</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_2_1_5_2"><b>suppose </b></a><a NAME="E1:29_1_1_2_1_5_2"/><i><font color="Green">E98</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font>
 ;<br/><div class="add"><a NAME="E2:29_1_1_2_1_5_2"/><b>then </b><i><font color="Green">E99</font></i>: 
<font color="Maroon">B'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="setwiseo.html#T14" target="_self">Th14</a>, </i>;<br/><a NAME="E3:29_1_1_2_1_5_2"/><i><font color="Green">E100</font></i>: 
for <font color="Olive">C</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font>  holds <br/>(  not <font color="Olive">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or  not <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> or  not <font color="Maroon">B'</font> <a href="hidden.html#R1">=</a> <font color="Olive">C</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="txt" href="setwiseo.html#E17"><i><font color="Green">Lemma32</font></i></a>, </i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_2_1_5_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1_1_2_1_5_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x'</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_2_1_5_2_1_1"><b>suppose </b></a><a NAME="E1:29_1_1_2_1_5_2_1_1"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x'</font>
 ;<br/><div class="add"><a NAME="E2:29_1_1_2_1_5_2_1_1"/><b>then </b>
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="setwiseo.html#T16" target="_self">Th16</a>, </i>;<br/><a NAME="E3:29_1_1_2_1_5_2_1_1"/><b>then </b><i><font color="Green">E102</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="setwiseo.html#K6" target="_self">\</a> <font color="Maroon">B'</font>
 <b>by </b><i><a class="ref" href="setwiseo.html#T16" target="_self">Th16</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E4:29_1_1_2_1_5_2_1_1"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x'</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="setwiseo.html#E17"><i><font color="Green">Lemma32</font></i></a>, , </i>;<br/><a NAME="E5:29_1_1_2_1_5_2_1_1"/><b>then </b>
for <font color="Olive">C</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font>  holds <br/>(  not <font color="Olive">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or  not <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> or  not <font color="Maroon">B'</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x'</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Olive">C</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i/>;<br/><b>hence </b><font color="Maroon">J</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">B'</font> <a href="setwiseo.html#K5" target="_self">\/</a> <span class="p2"><a href="setwiseo.html#K2" target="_self">{.</a><span class="default"><font color="Maroon">x'</font></span><a href="setwiseo.html#K2" target="_self">.}</a></span></span>)</span> = 
<font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">B'</font> <a href="setwiseo.html#K5" target="_self">\/</a> <span class="p2"><a href="setwiseo.html#K2" target="_self">{.</a><span class="default"><font color="Maroon">x'</font></span><a href="setwiseo.html#K2" target="_self">.}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="setwiseo.html#T13" target="_self">Th13</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="setwiseo.html#T18" target="_self">Th18</a>, </i>
<br/>.= 
<font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font></span>)</span>
<b>by </b><i><a class="ref" href="setwiseo.html#T13" target="_self">Th13</a>, <a class="ref" href="setwiseo.html#T19" target="_self">Th19</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_2_1_5_2_1_2"><b>suppose </b></a><a NAME="E1:29_1_1_2_1_5_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font>
 ;<br/><div class="add"><b>hence </b><font color="Maroon">J</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">B'</font> <a href="setwiseo.html#K5" target="_self">\/</a> <span class="p2"><a href="setwiseo.html#K2" target="_self">{.</a><span class="default"><font color="Maroon">x'</font></span><a href="setwiseo.html#K2" target="_self">.}</a></span></span>)</span> = 
<font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font></span>)</span>
<b>by </b><i><a class="ref" href="setwiseo.html#T13" target="_self">Th13</a>, , <a class="ref" href="setwiseo.html#T18" target="_self">Th18</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font></span>)</span>
<b>by </b><i><a class="ref" href="setwiseo.html#T13" target="_self">Th13</a>, <a class="ref" href="setwiseo.html#T19" target="_self">Th19</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E3:29_1_1"/>
for <font color="Olive">B</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font>]
 
<b>from </b><i><a class="ref" href="setwiseo.html#S2" target="_self">SETWISEO:sch 2</a>(, <a class="txt" href="setwiseo.html#E1:29_1_1"><i><font color="Green">E21</font></i></a>);<br/></i>
<b>then consider </b><font color="Maroon">G</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font><b> such that </b><br/><a NAME="E5:29_1_1"/><i><font color="Green">E103</font></i>: 
( ( for <font color="Olive">e</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>  st <font color="Olive">e</font> <a href="binop_1.html#R3">is_a_unity_wrt</a> <font color="Maroon">F</font> holds <br/><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Olive">e</font> ) &amp; ( for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> holds  <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> ) &amp; ( for <font color="Olive">B'</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font>  st <font color="Olive">B'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> &amp; <font color="Olive">B'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/>for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="setwiseo.html#K6" target="_self">\</a> <font color="Olive">B'</font> holds <br/><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">B'</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Olive">B'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> ) )
 ;<br/>
<b>take </b>
<font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B</font>
;<br/>

<b>take </b>
<font color="Maroon">G</font>
;<br/>

<b>thus </b><a NAME="E6:29_1_1"/>
( <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B</font> &amp; ( for <font color="Olive">e</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>  st <font color="Olive">e</font> <a href="binop_1.html#R3">is_a_unity_wrt</a> <font color="Maroon">F</font> holds <br/><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Olive">e</font> ) &amp; ( for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> holds  <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> ) &amp; ( for <font color="Olive">B'</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font>  st <font color="Olive">B'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> &amp; <font color="Olive">B'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/>for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="setwiseo.html#K6" target="_self">\</a> <font color="Olive">B'</font> holds <br/><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">B'</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Olive">B'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> ) )
 <b>by </b><i/>;<br/>


</div>
