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

<b>let </b><font color="Maroon">Y</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">PA</font> be    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">Y</font>;<br/><b>let </b><font color="Maroon">F</font> be   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Y</font>;<br/>





<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:11_1"/>
( <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">F</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="11_1_1"><b>suppose </b></a><a NAME="E1:11_1_1"/>
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:11_1_1"/><b>then </b>
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>;<br/><b>hence </b><a NAME="E3:11_1_1"/>
(  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; ( for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> holds <br/><font color="Olive">X</font> <a href="partit1.html#R1" target="_self">is_a_dependent_set_of</a> <font color="Maroon">PA</font> ) implies  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">F</font> <a href="partit1.html#R1" target="_self">is_a_dependent_set_of</a> <font color="Maroon">PA</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_2"><b>suppose </b></a><a NAME="E1:11_1_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">F'</font> = <font color="Maroon">F</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Y</font> ;<br/><b>assume </b><a NAME="E3:11_1_2"/><i><font color="Green">E43</font></i>: 
(  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; ( for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> holds <br/><font color="Olive">X</font> <a href="partit1.html#R1" target="_self">is_a_dependent_set_of</a> <font color="Maroon">PA</font> ) )
 ;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>( <font color="Maroon">a<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">PA</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">a<sub>2</sub></font> );<br/><a NAME="E4:11_1_2"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> holds <br/> ex <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">X</font>,<font color="Olive">B</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_2_1"><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="E1:11_1_2_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<a NAME="E2:11_1_2_1"/><b>then </b>
<font color="Maroon">X</font> <a href="partit1.html#R1" target="_self">is_a_dependent_set_of</a> <font color="Maroon">PA</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:11_1_2_1"/>
 ex <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">PA</font> &amp; <font color="Olive">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Olive">B</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:11_1_2"/><i><font color="Green">E48</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> &amp; ( for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">X</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">X</font>] ) )
 <b>from </b><i><a class="ref" href="zfrefle1.html#S1">ZFREFLE1:sch 1</a>();<br/></i><a NAME="E7:11_1_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Y</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:11_1_2_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:11_1_2_2"/><i><font color="Green">E52</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">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> <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="E4:11_1_2_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">PA</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:11_1_2_2"/><b>then </b>
<font color="Maroon">y</font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E6:11_1_2_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Y</font></span>)</span>
 ;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">F'</font>, <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Y</font></span>)</span> <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">F'</font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Y</font></span>)</span> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/><b>reconsider </b><font color="Maroon">T</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">X</font>) where <font color="Olive">X</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">F'</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">X</font>] </span>}</span>  as   <a href="setfam_1.html#NM1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Y</font></span>)</span> <b>from </b><i><a class="ref" href="domain_1.html#S8">DOMAIN_1:sch 8</a>();<br/></i><b>reconsider </b><font color="Maroon">T</font> = <font color="Maroon">T</font> as   <a href="setfam_1.html#NM1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Y</font></span>)</span> ;<br/><b>take </b><font color="Maroon">B</font> =  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="partit1.html#D1" target="_self">PARTIT1:def 1</a><br/><b>thus </b><a NAME="E11:11_1_2"/>
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">PA</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:11_1_2_3"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>

<b>consider </b><font color="Maroon">X</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:11_1_2_3"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E4:11_1_2_3"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:11_1_2_3"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</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="setfam_1.html#T58">SETFAM_1:58</a></i>;<br/>
<a NAME="E6:11_1_2_3"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">PA</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E7:11_1_2_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E12:11_1_2"/>
<font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_2_4"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">X</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:11_1_2_4"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E3:11_1_2_4"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font>
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">A</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:11_1_2_4"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">A</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">AA</font> =  <a href="t_1topsp.html#K1">EqClass</a> <font color="Maroon">A</font>,<font color="Maroon">PA</font>;<br/>
<a NAME="E7:11_1_2_4"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">F</font>
 
<b>by </b><i>, , <a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>;<br/>
<a NAME="E8:11_1_2_4"/>
for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> holds <br/> <a href="t_1topsp.html#K1">EqClass</a> <font color="Maroon">A</font>,<font color="Maroon">PA</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_2_4_1"><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="E1:11_1_2_4_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font>
 ;<br/>

<b>then consider </b><font color="Maroon">z</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">F'</font><b> such that </b><br/><a NAME="E3:11_1_2_4_1"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 ;<br/>
<a NAME="E4:11_1_2_4_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">PA</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:11_1_2_4_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E6:11_1_2_4_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">z</font>
 
<b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<a NAME="E7:11_1_2_4_1"/><b>then </b>
 ex <font color="Olive">A0</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Olive">A0</font> &amp; <font color="Olive">A0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> )
 
<b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E8:11_1_2_4_1"/>
 <a href="t_1topsp.html#K1">EqClass</a> <font color="Maroon">A</font>,<font color="Maroon">PA</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, , <a class="ref" href="t_1topsp.html#D1">T_1TOPSP:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:11_1_2_4"/><b>then </b>
 <a href="t_1topsp.html#K1">EqClass</a> <font color="Maroon">A</font>,<font color="Maroon">PA</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">T</font>
 
<b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E10:11_1_2_4"/>
<font color="Maroon">B</font> <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="setfam_1.html#D10">SETFAM_1:def 10</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E13:11_1_2"/>
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_2_5"><b>proof </b></a><div class="add">

<a NAME="E1:11_1_2_5"/><i><font color="Green">E79</font></i>: 
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_2_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:11_1_2_5_1"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">F</font>
 ;<br/>

<a NAME="E2:11_1_2_5_1"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">PA</font> = <font color="Maroon">PA</font> as    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">Y</font> ;<br/>
<b>set </b><font color="Maroon">A</font> =  <a href="t_1topsp.html#K1">EqClass</a> <font color="Maroon">x</font>,<font color="Maroon">PA</font>;<br/>
<b>consider </b><font color="Maroon">X</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:11_1_2_5_1"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E7:11_1_2_5_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font>
 
<b>by </b><i/>;<br/>
<a NAME="E8:11_1_2_5_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="t_1topsp.html#K1">EqClass</a> <font color="Maroon">x</font>,<font color="Maroon">PA</font>
 
<b>by </b><i><a class="ref" href="t_1topsp.html#D1">T_1TOPSP:def 1</a></i>;<br/>
<a NAME="E9:11_1_2_5_1"/>
for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> holds <br/> <a href="t_1topsp.html#K1">EqClass</a> <font color="Maroon">x</font>,<font color="Maroon">PA</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_2_5_1_1"><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="E1:11_1_2_5_1_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font>
 ;<br/>

<b>then consider </b><font color="Maroon">z</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">F'</font><b> such that </b><br/><a NAME="E3:11_1_2_5_1_1"/><i><font color="Green">E85</font></i>: 
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 ;<br/>
<a NAME="E4:11_1_2_5_1_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">PA</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:11_1_2_5_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E6:11_1_2_5_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<a NAME="E7:11_1_2_5_1_1"/><b>then </b>
 ex <font color="Olive">A0</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">A0</font> &amp; <font color="Olive">A0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E8:11_1_2_5_1_1"/>
 <a href="t_1topsp.html#K1">EqClass</a> <font color="Maroon">x</font>,<font color="Maroon">PA</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, , <a class="ref" href="t_1topsp.html#D1">T_1TOPSP:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:11_1_2_5_1"/><b>then </b>
 <a href="t_1topsp.html#K1">EqClass</a> <font color="Maroon">x</font>,<font color="Maroon">PA</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">T</font>
 
<b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<a NAME="E11:11_1_2_5_1"/><b>then </b>
 <a href="t_1topsp.html#K1">EqClass</a> <font color="Maroon">x</font>,<font color="Maroon">PA</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">T</font>
 
<b>by </b><i>, <a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>;<br/>
<b>hence </b><a NAME="E12:11_1_2_5_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:11_1_2_5"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_2_5_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:11_1_2_5_2"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">B</font>
 ;<br/>

<b>consider </b><font color="Maroon">X</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:11_1_2_5_2"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E4:11_1_2_5_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font>
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:11_1_2_5_2"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E7:11_1_2_5_2"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">T</font>
 
<b>by </b><i>, , <a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>;<br/>
<a NAME="E8:11_1_2_5_2"/>
for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> holds <br/><font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_2_5_2_1"><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="E1:11_1_2_5_2_1"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<a NAME="E2:11_1_2_5_2_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font>
 
;<br/>
<a NAME="E3:11_1_2_5_2_1"/><b>then </b><i><font color="Green">E93</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</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="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<a NAME="E4:11_1_2_5_2_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:11_1_2_5_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:11_1_2_5_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E10:11_1_2_5_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:11_1_2_5"/>
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


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

</div>
