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

<b>let </b><font color="Maroon">C</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">C'</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><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">C</font>;<br/><b>let </b><font color="Maroon">A</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">C'</font>;<br/><b>let </b><font color="Maroon">F</font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">C</font>,<font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">C'</font>,<font color="Maroon">D</font>;<br/>















<b>assume </b><a NAME="E1:6"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">F</font> is <a href="binop_1.html#V1">commutative</a> &amp; <font color="Maroon">F</font> is <a href="binop_1.html#V2">associative</a> )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">C'</font>] <b>means </b>( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">F</font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a>  ) implies 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">C</font>  st  ex <font color="Olive">s</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>(  <a href="relat_1.html#K1">dom</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> &amp; <font color="Olive">s</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Olive">s</font> ) holds <br/><font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <font color="Olive">B</font>,<font color="Maroon">f</font> );<br/>
<a NAME="E2:6"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="setwiseo.html#K1">{}.</a> <font color="Maroon">C'</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_1"/><i><font color="Green">E55</font></i>: 
(  <a href="setwiseo.html#K1">{}.</a> <font color="Maroon">C'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">F</font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a>  )
 ;<br/>

<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">C</font>;<br/>

<b>given </b><font color="Maroon">s</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><a NAME="E2:6_1"/><i><font color="Green">E56</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="setwiseo.html#K1">{}.</a> <font color="Maroon">C'</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> &amp; <font color="Maroon">s</font> is <a href="funct_1.html#V2">one-to-one</a> )
 <b>and </b><br/><a NAME="E3:6_1"/>
<font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K1">{}.</a> <font color="Maroon">C'</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">s</font>
 ;<br/>

<a NAME="E4:6_1"/>
( <font color="Maroon">B</font>, <a href="xboole_0.html#K1">{}</a>  <a href="tarski.html#R2">are_equipotent</a>  &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="setwiseo.html#K1">{}.</a> <font color="Maroon">C</font> )
 
<b>by </b><i>, <a class="ref" href="wellord2.html#D4">WELLORD2:def 4</a></i>;<br/>
<a NAME="E5:6_1"/><b>then </b>
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="setwiseo.html#K1">{}.</a> <font color="Maroon">C</font> &amp; <font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K1">{}.</a> <font color="Maroon">C'</font></span>)</span>,<font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">F</font> )
 
<b>by </b><i>, , <a class="ref" href="card_1.html#T46">CARD_1:46</a>, <a class="ref" href="setwiseo.html#T40">SETWISEO:40</a></i>;<br/>
<b>hence </b><a NAME="E6:6_1"/>
<font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K1">{}.</a> <font color="Maroon">C'</font></span>)</span>,<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">B</font>,<font color="Maroon">f</font>
 <b>by </b><i>, , <a class="ref" href="setwiseo.html#T40">SETWISEO:40</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:6"/><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">C'</font><br/> for <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C'</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="setwiseo.html#K5">\/</a> <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Olive">b</font></span><a href="setwiseo.html#K2">.}</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2"><b>proof </b></a><div class="add">

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



<b>assume </b><b>that </b><br/><a NAME="E1:6_2"/><i><font color="Green">E58</font></i>: 
( ( <font color="Maroon">A'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">F</font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a>  ) implies 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">C</font>  st  ex <font color="Olive">s</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>(  <a href="relat_1.html#K1">dom</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A'</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> &amp; <font color="Olive">s</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Olive">s</font> ) holds <br/><font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">A'</font>,<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <font color="Olive">B</font>,<font color="Maroon">f</font> )
 <b>and </b><br/><a NAME="E2:6_2"/><i><font color="Green">E60</font></i>: 
not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A'</font>
 ;<br/>

<b>set </b><font color="Maroon">A</font> = <font color="Maroon">A'</font> <a href="setwiseo.html#K5">\/</a> <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">a</font></span><a href="setwiseo.html#K2">.}</a></span>;<br/>
<b>assume </b><a NAME="E3:6_2"/>
( <font color="Maroon">A'</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a</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>  or <font color="Maroon">F</font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a>  )
 ;<br/>

<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">C</font>;<br/>

<b>given </b><font color="Maroon">s</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><a NAME="E4:6_2"/><i><font color="Green">E61</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A'</font> <a href="setwiseo.html#K5">\/</a> <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">a</font></span><a href="setwiseo.html#K2">.}</a></span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> &amp; <font color="Maroon">s</font> is <a href="funct_1.html#V2">one-to-one</a> )
 <b>and </b><br/><a NAME="E5:6_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">A'</font> <a href="setwiseo.html#K5">\/</a> <span class="p2"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">a</font></span><a href="setwiseo.html#K2">.}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">s</font>
 ;<br/>

<a NAME="E6:6_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A'</font> <a href="setwiseo.html#K5">\/</a> <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">a</font></span><a href="setwiseo.html#K2">.}</a></span>
 
<b>by </b><i><a class="ref" href="setwiseo.html#T6">SETWISEO:6</a></i>;<br/>
<a NAME="E7:6_2"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E8:6_2"/>
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="finsub_1.html#D5">FINSUB_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c</font> = <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font> <b>by </b><i/>;<br/>
<a NAME="E10:6_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C'</font>
 
;<br/>
<a NAME="E11:6_2"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E12:6_2"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A'</font> <a href="setwiseo.html#K5">\/</a> <span class="p2"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">a</font></span><a href="setwiseo.html#K2">.}</a></span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E13:6_2"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">s</font></span>)</span> &amp; <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T71">FUNCT_1:71</a>, <a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<a NAME="E14:6_2"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<b>set </b><font color="Maroon">B'</font> = <font color="Maroon">B</font> <a href="setwiseo.html#K6">\</a> <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">c</font></span><a href="setwiseo.html#K2">.}</a></span>;<br/>
<b>set </b><font color="Maroon">s'</font> = <font color="Maroon">s</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A'</font>;<br/>
<div><i><font color="Green">E68</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E1:6_2_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A'</font></span>)</span> implies <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="setwiseo.html#K6">\</a> <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">c</font></span><a href="setwiseo.html#K2">.}</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_2_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A'</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:6_2_1_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A'</font></span>)</span>
 <b>and </b><br/><a NAME="E4:6_2_1_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:6_2_1_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">s</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">A'</font>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<a NAME="E6:6_2_1_1"/><b>then </b><i><font color="Green">E73</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">s</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A'</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E7:6_2_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">s</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> )
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E8:6_2_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> )
 
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E9:6_2_1_1"/><b>then </b>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i>, , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E10:6_2_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="setwiseo.html#K6">\</a> <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">c</font></span><a href="setwiseo.html#K2">.}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:6_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="setwiseo.html#K6">\</a> <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">c</font></span><a href="setwiseo.html#K2">.}</a></span>
 ;<br/><a NAME="E3:6_2_1"/><b>then </b><i><font color="Green">E74</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:6_2_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E6:6_2_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E7:6_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> &amp; ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A'</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a</font></span><a href="tarski.html#K1">}</a></span> ) )
 <b>by </b><i>, , , , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E8:6_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">s</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">A'</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E9:6_2_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A'</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#T71">FUNCT_1:71</a>, <a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/><b>hence </b><a NAME="E10:6_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A'</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div>
<a NAME="E16:6_2"/>
<font color="Maroon">A'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A'</font> <a href="setwiseo.html#K5">\/</a> <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">a</font></span><a href="setwiseo.html#K2">.}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E17:6_2"/><b>then </b><i><font color="Green">E77</font></i>: 
( <font color="Maroon">s</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A'</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A'</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A'</font> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A'</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="setwiseo.html#K6">\</a> <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">c</font></span><a href="setwiseo.html#K2">.}</a></span> )
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T84">FUNCT_1:84</a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2_2"><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>thus </b><a NAME="E1:6_2_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A'</font></span>)</span> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A'</font></span>)</span></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_2_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A'</font></span>)</span>
 ;<br/>

<a NAME="E2:6_2_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">A'</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<a NAME="E3:6_2_2_1"/><b>then </b><i><font color="Green">E78</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A'</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:6_2_2_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A'</font> <a href="setwiseo.html#K5">\/</a> <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">a</font></span><a href="setwiseo.html#K2">.}</a></span> )
 
<b>by </b><i><a class="ref" href="setwiseo.html#T6">SETWISEO:6</a></i>;<br/>
<a NAME="E5:6_2_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A'</font> <a href="setwiseo.html#K5">\/</a> <span class="p2"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">a</font></span><a href="setwiseo.html#K2">.}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:6_2_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">s</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<a NAME="E7:6_2_2_1"/><b>then </b><i><font color="Green">E83</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">s</font> &amp; <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/>
<a NAME="E8:6_2_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">s</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">A'</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E9:6_2_2_1"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<a NAME="E10:6_2_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<b>hence </b><a NAME="E11:6_2_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A'</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:6_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A'</font></span>)</span></span>)</span>
 ;<br/><a NAME="E3:6_2_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A'</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><a NAME="E4:6_2_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">s</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">A'</font> &amp; <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#T70">FUNCT_1:70</a>, <a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/><a NAME="E5:6_2_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">s</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A'</font> &amp; <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E6:6_2_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A'</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">A'</font> <a href="setwiseo.html#K5">\/</a> <span class="p3"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">a</font></span><a href="setwiseo.html#K2">.}</a></span></span>)</span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><a NAME="E7:6_2_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A'</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A'</font> <a href="setwiseo.html#K5">\/</a> <span class="p2"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">a</font></span><a href="setwiseo.html#K2">.}</a></span></span>)</span> )
 <b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/><a NAME="E8:6_2_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A'</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E9:6_2_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">A'</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E10:6_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A'</font></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/></div><b>end;</b></div>
<a NAME="E19:6_2"/><b>then </b><i><font color="Green">E85</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A'</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A'</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<a NAME="E20:6_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>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A'</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A'</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2_3"><b>proof </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="E1:6_2_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A'</font></span>)</span>
 ;<br/>

<a NAME="E2:6_2_3"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">A'</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<a NAME="E3:6_2_3"/><b>then </b><i><font color="Green">E86</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A'</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:6_2_3"/><b>then </b><i><font color="Green">E87</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A'</font> <a href="setwiseo.html#K5">\/</a> <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">a</font></span><a href="setwiseo.html#K2">.}</a></span> )
 
<b>by </b><i><a class="ref" href="setwiseo.html#T6">SETWISEO:6</a></i>;<br/>
<a NAME="E5:6_2_3"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A'</font> <a href="setwiseo.html#K5">\/</a> <span class="p2"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">a</font></span><a href="setwiseo.html#K2">.}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:6_2_3"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">s</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<a NAME="E7:6_2_3"/><b>then </b><i><font color="Green">E88</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">s</font> &amp; <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/>
<a NAME="E8:6_2_3"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">s</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">A'</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E9:6_2_3"/><b>then </b><i><font color="Green">E89</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<a NAME="E10:6_2_3"/><b>then </b><i><font color="Green">E90</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A'</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E21:6_2"/><b>then </b><i><font color="Green">E91</font></i>: 
<font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A'</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<a NAME="E22:6_2"/>
<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="setwiseo.html#K6">\</a> <span class="p2"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">c</font></span><a href="setwiseo.html#K2">.}</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">c</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</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">c</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/>
<a NAME="E23:6_2"/><b>then </b><i><font color="Green">E92</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="setwiseo.html#K6">\</a> <span class="p2"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">c</font></span><a href="setwiseo.html#K2">.}</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">c</font></span><a href="tarski.html#K1">}</a></span> &amp; not <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="setwiseo.html#K6">\</a> <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">c</font></span><a href="setwiseo.html#K2">.}</a></span> )
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T46">ZFMISC_1:46</a>, <a class="ref" href="zfmisc_1.html#T64">ZFMISC_1:64</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E1:6_2_4"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g</font> <a href="setwiseo.html#K8">.:</a> <font color="Maroon">A'</font> implies <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="setwiseo.html#K8">.:</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="setwiseo.html#K6">\</a> <span class="p2"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">c</font></span><a href="setwiseo.html#K2">.}</a></span></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2_4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_2_4_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g</font> <a href="setwiseo.html#K8">.:</a> <font color="Maroon">A'</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:6_2_4_1"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E4:6_2_4_1"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A'</font>
 <b>and </b><br/><a NAME="E5:6_2_4_1"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E6:6_2_4_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">A'</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E7:6_2_4_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A'</font></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A'</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a>, <a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<a NAME="E8:6_2_4_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A'</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<a NAME="E9:6_2_4_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="setwiseo.html#K6">\</a> <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">c</font></span><a href="setwiseo.html#K2">.}</a></span> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E10:6_2_4_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="setwiseo.html#K8">.:</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="setwiseo.html#K6">\</a> <span class="p2"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">c</font></span><a href="setwiseo.html#K2">.}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:6_2_4"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="setwiseo.html#K8">.:</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="setwiseo.html#K6">\</a> <span class="p2"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">c</font></span><a href="setwiseo.html#K2">.}</a></span></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:6_2_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E5:6_2_4"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="setwiseo.html#K6">\</a> <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">c</font></span><a href="setwiseo.html#K2">.}</a></span>
 <b>and </b><br/><a NAME="E6:6_2_4"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><b>set </b><font color="Maroon">x'</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A'</font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>;<br/><a NAME="E7:6_2_4"/><i><font color="Green">E99</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A'</font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A'</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T54">FUNCT_1:54</a></i>;<br/><a NAME="E8:6_2_4"/>
<font color="Maroon">A'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C'</font>
 <b>by </b><i><a class="ref" href="finsub_1.html#D5">FINSUB_1:def 5</a></i>;<br/><a NAME="E9:6_2_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A'</font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C'</font>
 <b>by </b><i/>;<br/><a NAME="E10:6_2_4"/><b>then </b><i><font color="Green">E100</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A'</font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E11:6_2_4"/>
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A'</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A'</font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T57">FUNCT_1:57</a></i>;<br/><b>then </b><font color="Maroon">y</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A'</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A'</font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A'</font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/><b>hence </b><a NAME="E13:6_2_4"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g</font> <a href="setwiseo.html#K8">.:</a> <font color="Maroon">A'</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/></div><b>end;</b></div>
<a NAME="E25:6_2"/><b>then </b><i><font color="Green">E101</font></i>: 
<font color="Maroon">f</font> <a href="setwiseo.html#K8">.:</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="setwiseo.html#K6">\</a> <span class="p2"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">c</font></span><a href="setwiseo.html#K2">.}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="setwiseo.html#K8">.:</a> <font color="Maroon">A'</font>
 
<b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2_5"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_2_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:6_2_5_1"/>
( <font color="Maroon">A'</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">A'</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="6_2_5_1_1"><b>suppose </b></a><a NAME="E1:6_2_5_1_1"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">A'</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:6_2_5_1_1"/>
<font color="Maroon">B</font> <a href="setwiseo.html#K6">\</a> <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">c</font></span><a href="setwiseo.html#K2">.}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="finsub_1.html#D5">FINSUB_1:def 5</a></i>;<br/><a NAME="E3:6_2_5_1_1"/><b>then </b>
( <font color="Maroon">B</font> <a href="setwiseo.html#K6">\</a> <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">c</font></span><a href="setwiseo.html#K2">.}</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">g</font> <a href="setwiseo.html#K8">.:</a> <font color="Maroon">A'</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relat_1.html#T149">RELAT_1:149</a></i>;<br/><a NAME="E4:6_2_5_1_1"/><b>then </b><i><font color="Green">E103</font></i>: 
<font color="Maroon">B</font> <a href="setwiseo.html#K6">\</a> <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">c</font></span><a href="setwiseo.html#K2">.}</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="relat_1.html#T152">RELAT_1:152</a></i>;<br/><b>thus </b><font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <span class="p1">(<span class="default"><font color="Maroon">A'</font> <a href="setwiseo.html#K5">\/</a> <span class="p2"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">a</font></span><a href="setwiseo.html#K2">.}</a></span></span>)</span>,<font color="Maroon">g</font> = 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font>
<b>by </b><i>, , , <a class="ref" href="setwiseo.html#T26">SETWISEO:26</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">B</font>,<font color="Maroon">f</font>
<b>by </b><i>, , , <a class="ref" href="setwiseo.html#T26">SETWISEO:26</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_2_5_1_2"><b>suppose </b></a><a NAME="E1:6_2_5_1_2"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">A'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:6_2_5_1_2"/>
<font color="Maroon">A'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C'</font>
 <b>by </b><i><a class="ref" href="finsub_1.html#D5">FINSUB_1:def 5</a></i>;<br/><a NAME="E3:6_2_5_1_2"/><b>then </b>
<font color="Maroon">A'</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E4:6_2_5_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="setwiseo.html#K8">.:</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="setwiseo.html#K6">\</a> <span class="p2"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">c</font></span><a href="setwiseo.html#K2">.}</a></span></span>)</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="relat_1.html#T152">RELAT_1:152</a></i>;<br/><a NAME="E5:6_2_5_1_2"/><b>then </b><i><font color="Green">E105</font></i>: 
<font color="Maroon">B</font> <a href="setwiseo.html#K6">\</a> <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">c</font></span><a href="setwiseo.html#K2">.}</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="relat_1.html#T149">RELAT_1:149</a></i>;<br/><b>thus </b><font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <span class="p1">(<span class="default"><font color="Maroon">A'</font> <a href="setwiseo.html#K5">\/</a> <span class="p2"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">a</font></span><a href="setwiseo.html#K2">.}</a></span></span>)</span>,<font color="Maroon">g</font> = 
<font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">A'</font>,<font color="Maroon">g</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</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">F</font> <a href="setwiseo.html#K7">$$</a> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="setwiseo.html#K6">\</a> <span class="p3"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">c</font></span><a href="setwiseo.html#K2">.}</a></span></span>)</span>,<font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i>, , , , </i>
<br/>.= 
<font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">B</font>,<font color="Maroon">f</font>
<b>by </b><i>, , , </i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E27:6_2"/>
<font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <span class="p1">(<span class="default"><font color="Maroon">A'</font> <a href="setwiseo.html#K5">\/</a> <span class="p2"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">a</font></span><a href="setwiseo.html#K2">.}</a></span></span>)</span>,<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">B</font>,<font color="Maroon">f</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:6"/>
for <font color="Olive">A</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">C'</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">A</font>]
 
<b>from </b><i><a class="ref" href="setwiseo.html#S2">SETWISEO:sch 2</a>(, );<br/></i>
<b>hence </b><a NAME="E5:6"/>
( ( not <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; not <font color="Maroon">F</font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a>  ) or for <font color="Olive">s</font> being  <a href="funct_1.html#NM1">Function</a>  holds <br/>(  not  <a href="relat_1.html#K1">dom</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> or  not  <a href="relat_1.html#K2">rng</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> or  not <font color="Olive">s</font> is <a href="funct_1.html#V2">one-to-one</a> or  not <font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Olive">s</font> ) or <font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">A</font>,<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">B</font>,<font color="Maroon">f</font> )
 ;<br/>


</div>
