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

<b>let </b><font color="Maroon">X</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">A</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>, <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">A</font>;<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">X</font>] <b>means </b> <a href="setwiseo.html#K10" target="_self">FinUnion</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">f</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="setwiseo.html#K8" target="_self">.:</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E1:66"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="setwiseo.html#K1" target="_self">{}.</a> <font color="Maroon">X</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><b>proof </b></a><div class="add">

<b>thus </b> <a href="setwiseo.html#K10" target="_self">FinUnion</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K1" target="_self">{}.</a> <font color="Maroon">X</font></span>)</span>,<font color="Maroon">f</font> = 
 <a href="tarski.html#K3">union</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T2">ZFMISC_1:2</a></i>
<br/>.= 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="setwiseo.html#K8" target="_self">.:</a> <span class="p2">(<span class="default"><a href="setwiseo.html#K1" target="_self">{}.</a> <font color="Maroon">X</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T149">RELAT_1:149</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E2:66"/><i><font color="Green">E15</font></i>: 
for <font color="Olive">B</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font><br/> for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font> <a href="setwiseo.html#K5" target="_self">\/</a> <span class="p1"><a href="setwiseo.html#K2" target="_self">{.</a><span class="default"><font color="Olive">i</font></span><a href="setwiseo.html#K2" target="_self">.}</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2"><b>proof </b></a><div class="add">

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



<b>assume </b><a NAME="E1:66_2"/>
 <a href="setwiseo.html#K10" target="_self">FinUnion</a> <font color="Maroon">B</font>,<font color="Maroon">f</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="setwiseo.html#K8" target="_self">.:</a> <font color="Maroon">B</font></span>)</span>
 ;<br/>

<b>hence </b> <a href="setwiseo.html#K10" target="_self">FinUnion</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="setwiseo.html#K5" target="_self">\/</a> <span class="p2"><a href="setwiseo.html#K2" target="_self">{.</a><span class="default"><font color="Maroon">i</font></span><a href="setwiseo.html#K2" target="_self">.}</a></span></span>)</span>,<font color="Maroon">f</font> = 
<span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="setwiseo.html#K8" target="_self">.:</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="setwiseo.html#K8" target="_self">.:</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="zfmisc_1.html#T31">ZFMISC_1:31</a></i>
<br/>.= 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="setwiseo.html#K8" target="_self">.:</a> <font color="Maroon">B</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="zfmisc_1.html#T96">ZFMISC_1:96</a></i>
<br/>.= 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="setwiseo.html#K8" target="_self">.:</a> <font color="Maroon">B</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T153">RELAT_1:153</a></i>
;<br/><br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:66"/>
for <font color="Olive">B</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font>]
 <b>from </b><i><a class="ref" href="setwiseo.html#S4" target="_self">SETWISEO:sch 4</a>(<a class="ref" href="setwiseo.html#T22" target="_self">Th22</a>, <a class="ref" href="setwiseo.html#T23" target="_self">Th23</a>);</i><br/>


</div>
