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

<b>let </b><font color="Maroon">Omega</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">Sigma</font> be   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">Omega</font>;<br/><b>let </b><font color="Maroon">P</font> be    <a href="prob_1.html#M3">Probability</a> of <font color="Maroon">Sigma</font>;<br/><b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Omega</font>;<br/>







<b>assume </b><a NAME="E1:48"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">A</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> iff  ex <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sigma</font> &amp;  ex <font color="Olive">C</font> being   <a href="prob_4.html#M1" target="_self">thin</a> of <font color="Maroon">P</font> st <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">C</font> ) )
 ;<br/>

<a NAME="E2:48"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">A</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Omega</font>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> holds <br/><font color="Olive">A</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Omega</font>;<br/>

<b>assume </b><a NAME="E1:48_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/>

<a NAME="E2:48_1"/>
 ex <font color="Olive">Q</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">Q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sigma</font> &amp;  ex <font color="Olive">W</font> being   <a href="prob_4.html#M1" target="_self">thin</a> of <font color="Maroon">P</font> st <font color="Maroon">Omega</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">Q</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">W</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">B</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:48_1_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sigma</font> &amp;  ex <font color="Olive">C</font> being   <a href="prob_4.html#M1" target="_self">thin</a> of <font color="Maroon">P</font> st <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">C</font> )
 <b>by </b><i><a class="ref" href="prob_4.html#T2" target="_self">Th2</a>, <a class="txt" href="prob_4.html#E1:48_1"><i><font color="Green">E39</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">C</font> being    <a href="prob_4.html#M1" target="_self">thin</a> of <font color="Maroon">P</font><b> such that </b><br/><a NAME="E4:48_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">C</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">G</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:48_1_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">G</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sigma</font> &amp; <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="txt" href="prob_4.html#E1:48"><i><font color="Green">E24</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">B</font> = <font color="Maroon">B</font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Omega</font> <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">H</font> = <font color="Maroon">Omega</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">B</font>;<br/>
<a NAME="E8:48_1_1"/>
<font color="Maroon">B</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">Sigma</font>
 
<b>by </b><i>, <a class="ref" href="prob_1.html#D1">PROB_1:def 1</a></i>;<br/>
<a NAME="E9:48_1_1"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">Omega</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sigma</font>
 
;<br/>
<a NAME="E10:48_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">Omega</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">Omega</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">B</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">C</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T41">XBOOLE_1:41</a></i>;<br/>
<b>set </b><font color="Maroon">Q</font> = <span class="p1">(<span class="default"><font color="Maroon">Omega</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">B</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">G</font>;<br/>
<a NAME="E11:48_1_1"/><i><font color="Green">E54</font></i>: 
 ex <font color="Olive">W</font> being   <a href="prob_4.html#M1" target="_self">thin</a> of <font color="Maroon">P</font> st <font color="Maroon">Omega</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Omega</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">B</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Olive">W</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">W</font> = <span class="p1">(<span class="default"><font color="Maroon">Omega</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">B</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">C</font></span>)</span>;<br/>
<a NAME="E1:48_1_1_1"/><i><font color="Green">E55</font></i>: 
 ex <font color="Olive">R</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">R</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sigma</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">Omega</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">B</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">C</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Olive">R</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Olive">R</font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i/>;<br/>
<a NAME="E2:48_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">Omega</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">B</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">C</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">Omega</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">W</font> = <span class="p1">(<span class="default"><font color="Maroon">Omega</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">B</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">C</font></span>)</span> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Omega</font> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">W</font> = <font color="Maroon">W</font> as    <a href="prob_4.html#M1" target="_self">thin</a> of <font color="Maroon">P</font> <b>by </b><i>, <a class="txt" href="prob_4.html#E1:48"><i><font color="Green">E24</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">W</font>
;<br/>

<b>thus </b><a NAME="E5:48_1_1_1"/>
<font color="Maroon">Omega</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Omega</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">B</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">W</font>
 <b>by </b><i>, <a class="ref" href="prob_4.html#T3" target="_self">Th3</a>, </i>;<br/>


</div><b>end;</b></div>
<b>take </b>
<span class="p1">(<span class="default"><font color="Maroon">Omega</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">B</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">G</font>
;<br/>

<b>thus </b><a NAME="E12:48_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">Omega</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">B</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">G</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sigma</font> &amp;  ex <font color="Olive">W</font> being   <a href="prob_4.html#M1" target="_self">thin</a> of <font color="Maroon">P</font> st <font color="Maroon">Omega</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Omega</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">B</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">G</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Olive">W</font> )
 <b>by </b><i>, , , <a class="ref" href="prob_1.html#T44">PROB_1:44</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:48_1"/><b>then </b>
<font color="Maroon">Omega</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="prob_4.html#T2" target="_self">Th2</a></i>;<br/>
<b>hence </b><a NAME="E4:48_1"/>
<font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:48"/>
for <font color="Olive">A1</font> being  <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">Omega</font>  st ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">A1</font> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> ) holds <br/> <a href="prob_1.html#K2">Union</a> <font color="Olive">A1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A1</font> be   <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">Omega</font>;<br/>

<b>assume </b><a NAME="E1:48_2"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">A1</font> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">F</font> = <font color="Maroon">A1</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Omega</font> ;<br/>
<a NAME="E3:48_2"/><i><font color="Green">E78</font></i>: 
 <a href="prob_1.html#K2">Union</a> <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">A1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_3.html#D4">CARD_3:def 4</a></i>;<br/>
<a NAME="E4:48_2"/><i><font color="Green">E91</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:48_2"/><i><font color="Green">E92</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sigma</font> &amp;  ex <font color="Olive">C</font> being   <a href="prob_4.html#M1" target="_self">thin</a> of <font color="Maroon">P</font> st <font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">C</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:48_2_1"/>
<font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:48_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="hidden.html#R2">in</a> <font color="Maroon">Sigma</font> &amp;  ex <font color="Olive">C</font> being   <a href="prob_4.html#M1" target="_self">thin</a> of <font color="Maroon">P</font> st <font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">C</font> )
 <b>by </b><i><a class="ref" href="prob_4.html#T2" target="_self">Th2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:48_2"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="prob_4.html#K4" target="_self">COM</a> <font color="Maroon">Sigma</font>,<font color="Maroon">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:48_2_2"/>
 ex <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sigma</font> &amp;  ex <font color="Olive">C</font> being   <a href="prob_4.html#M1" target="_self">thin</a> of <font color="Maroon">P</font> st <font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">C</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:48_2_2"/>
<font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="prob_4.html#K4" target="_self">COM</a> <font color="Maroon">Sigma</font>,<font color="Maroon">P</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E7:48_2"/><b>then </b>
for <font color="Olive">n</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="prob_4.html#K4" target="_self">COM</a> <font color="Maroon">Sigma</font>,<font color="Maroon">P</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="prob_4.html#K4" target="_self">COM</a> <font color="Maroon">Sigma</font>,<font color="Maroon">P</font> <b>by </b><i>, <a class="ref" href="funct_2.html#T5">FUNCT_2:5</a></i>;<br/>
<b>consider </b><font color="Maroon">BSeq</font> being    <a href="prob_1.html#M1">SetSequence</a> of <font color="Maroon">Sigma</font><b> such that </b><br/><a NAME="E10:48_2"/><i><font color="Green">E94</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">BSeq</font> <a href="prob_4.html#K1" target="_self">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="prob_4.html#K6" target="_self">ProbPart</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span>
 <b>by </b><i/>;<br/>
<a NAME="E11:48_2"/><i><font color="Green">E95</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <font color="Maroon">BSeq</font> <a href="prob_4.html#K1" target="_self">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sigma</font> &amp; <font color="Maroon">BSeq</font> <a href="prob_4.html#K1" target="_self">.</a> <font color="Olive">n</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">BSeq</font> <a href="prob_4.html#K1" target="_self">.</a> <font color="Olive">n</font></span>)</span> is    <a href="prob_4.html#M1" target="_self">thin</a> of <font color="Maroon">P</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:48_2_3"/>
<font color="Maroon">BSeq</font> <a href="prob_4.html#K1" target="_self">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="prob_4.html#K6" target="_self">ProbPart</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:48_2_3"/>
( <font color="Maroon">BSeq</font> <a href="prob_4.html#K1" target="_self">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sigma</font> &amp; <font color="Maroon">BSeq</font> <a href="prob_4.html#K1" target="_self">.</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">BSeq</font> <a href="prob_4.html#K1" target="_self">.</a> <font color="Maroon">n</font></span>)</span> is    <a href="prob_4.html#M1" target="_self">thin</a> of <font color="Maroon">P</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">CSeq</font> being   <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">Omega</font><b> such that </b><br/><a NAME="E13:48_2"/><i><font color="Green">E96</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">CSeq</font> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">BSeq</font> <a href="prob_4.html#K1" target="_self">.</a> <font color="Olive">n</font></span>)</span>
 <b>by </b><i/>;<br/>
<a NAME="E14:48_2"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">CSeq</font> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font> is    <a href="prob_4.html#M1" target="_self">thin</a> of <font color="Maroon">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:48_2_4"/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">BSeq</font> <a href="prob_4.html#K1" target="_self">.</a> <font color="Maroon">n</font></span>)</span> is    <a href="prob_4.html#M1" target="_self">thin</a> of <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="prob_4.html#T3" target="_self">Th3</a></i>;<br/>
<b>hence </b><a NAME="E2:48_2_4"/>
<font color="Maroon">CSeq</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font> is    <a href="prob_4.html#M1" target="_self">thin</a> of <font color="Maroon">P</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">DSeq</font> being    <a href="prob_1.html#M1">SetSequence</a> of <font color="Maroon">Sigma</font><b> such that </b><br/><a NAME="E16:48_2"/><i><font color="Green">E97</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <font color="Maroon">CSeq</font> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">DSeq</font> <a href="prob_4.html#K1" target="_self">.</a> <font color="Olive">n</font> &amp; <font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">DSeq</font> <a href="prob_4.html#K1" target="_self">.</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="prob_4.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E17:48_2"/>
 ex <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sigma</font> &amp;  ex <font color="Olive">C</font> being   <a href="prob_4.html#M1" target="_self">thin</a> of <font color="Maroon">P</font> st  <a href="prob_1.html#K2">Union</a> <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">C</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2_5"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">B</font> =  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">BSeq</font></span>)</span>;<br/>
<a NAME="E1:48_2_5"/><i><font color="Green">E98</font></i>: 
 <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">BSeq</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_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="E1:48_2_5_1"/>
<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"><a href="relset_1.html#K5">rng</a> <font color="Maroon">BSeq</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">Z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:48_2_5_1"/><i><font color="Green">E99</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">BSeq</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E4:48_2_5_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">BSeq</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">n</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:48_2_5_1"/><i><font color="Green">E100</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">BSeq</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">P</font> = <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>;<br/>
<a NAME="E8:48_2_5_1"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">BSeq</font> <a href="prob_4.html#K1" target="_self">.</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="prob_4.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E9:48_2_5_1"/>
 ex <font color="Olive">P</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">P</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">P</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2_5_1_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>
;<br/>

<b>thus </b><a NAME="E1:48_2_5_1_1"/>
( <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> )
 <b>by </b><i>, , , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:48_2_5_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:48_2_5"/><i><font color="Green">E102</font></i>: 
 ex <font color="Olive">C</font> being   <a href="prob_4.html#M1" target="_self">thin</a> of <font color="Maroon">P</font> st  <a href="prob_1.html#K2">Union</a> <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">BSeq</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Olive">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2_5_2"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">C</font> = <span class="p1">(<span class="default"><a href="prob_1.html#K2">Union</a> <font color="Maroon">A1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">BSeq</font></span>)</span></span>)</span>;<br/>
<i><font color="Green">E138</font></i>:  <a href="prob_1.html#K2">Union</a> <font color="Maroon">A1</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_1.html#K2">Union</a> <font color="Maroon">A1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">BSeq</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">BSeq</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T51">XBOOLE_1:51</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">BSeq</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_1.html#K2">Union</a> <font color="Maroon">A1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">BSeq</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<a NAME="E2:48_2_5_2"/>
<span class="p1">(<span class="default"><a href="prob_1.html#K2">Union</a> <font color="Maroon">A1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">BSeq</font></span>)</span></span>)</span> is    <a href="prob_4.html#M1" target="_self">thin</a> of <font color="Maroon">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2_5_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:48_2_5_2_2"/><i><font color="Green">E139</font></i>: 
<span class="p1">(<span class="default"><a href="prob_1.html#K2">Union</a> <font color="Maroon">A1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">BSeq</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">CSeq</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2_5_2_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> ; <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="E1:48_2_5_2_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="prob_1.html#K2">Union</a> <font color="Maroon">A1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">BSeq</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:48_2_5_2_2_1"/><b>then </b><i><font color="Green">E140</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font></span>)</span> &amp; not <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"><a href="relset_1.html#K5">rng</a> <font color="Maroon">BSeq</font></span>)</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">Z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:48_2_5_2_2_1"/><i><font color="Green">E141</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">n</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:48_2_5_2_2_1"/><i><font color="Green">E142</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="prob_4.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E8:48_2_5_2_2_1"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">BSeq</font> <a href="prob_4.html#K1" target="_self">.</a> <font color="Maroon">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2_5_2_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:48_2_5_2_2_1_1"/><i><font color="Green">E143</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">BSeq</font> <a href="prob_4.html#K1" target="_self">.</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E2:48_2_5_2_2_1_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">BSeq</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:48_2_5_2_2_1_1"/><b>then </b>
<font color="Maroon">BSeq</font> <a href="prob_4.html#K1" target="_self">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">BSeq</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E4:48_2_5_2_2_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:48_2_5_2_2_1"/><b>then </b><i><font color="Green">E144</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">BSeq</font> <a href="prob_4.html#K1" target="_self">.</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="prob_4.html#T4" target="_self">Th4</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E10:48_2_5_2_2_1"/>
 ex <font color="Olive">Z</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">Z</font> &amp; <font color="Olive">Z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">CSeq</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2_5_2_2_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:48_2_5_2_2_1_2"/><i><font color="Green">E145</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">CSeq</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>take </b>
<font color="Maroon">CSeq</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font>
;<br/>

<b>thus </b><a NAME="E2:48_2_5_2_2_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CSeq</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">CSeq</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">CSeq</font> )
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:48_2_5_2_2_1"/>
<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"><a href="relset_1.html#K5">rng</a> <font color="Maroon">CSeq</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:48_2_5_2_2"/>
 <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">CSeq</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">DSeq</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2_5_2_2_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="E1:48_2_5_2_2_2"/>
<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"><a href="relset_1.html#K5">rng</a> <font color="Maroon">CSeq</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">Z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:48_2_5_2_2_2"/><i><font color="Green">E146</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">CSeq</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E4:48_2_5_2_2_2"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">CSeq</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">n</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:48_2_5_2_2_2"/><i><font color="Green">E147</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">CSeq</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E8:48_2_5_2_2_2"/><i><font color="Green">E148</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CSeq</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">CSeq</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">DSeq</font> <a href="prob_4.html#K1" target="_self">.</a> <font color="Maroon">n</font> )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E9:48_2_5_2_2_2"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">DSeq</font>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E10:48_2_5_2_2_2"/><b>then </b>
<font color="Maroon">DSeq</font> <a href="prob_4.html#K1" target="_self">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">DSeq</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E11:48_2_5_2_2_2"/>
<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"><a href="relset_1.html#K5">rng</a> <font color="Maroon">DSeq</font></span>)</span>
 <b>by </b><i><a class="ref" href="prob_4.html#T4" target="_self">Th4</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:48_2_5_2_2"/><b>then </b><i><font color="Green">E149</font></i>: 
<span class="p1">(<span class="default"><a href="prob_1.html#K2">Union</a> <font color="Maroon">A1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">BSeq</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">DSeq</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E4:48_2_5_2_2"/>
for <font color="Olive">A</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">A</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">DSeq</font> holds <br/><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Olive">A</font> <a href="hidden.html#R1">=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2_5_2_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:48_2_5_2_2_3"/><i><font color="Green">E150</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">DSeq</font>
 ;<br/>

<a NAME="E2:48_2_5_2_2_3"/><i><font color="Green">E151</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">DSeq</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Sigma</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:48_2_5_2_2_3"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">DSeq</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:48_2_5_2_2_3"/><b>then </b><i><font color="Green">E152</font></i>: 
 ex <font color="Olive">n</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">DSeq</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">A</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Sigma</font> <b>by </b><i>, <a class="ref" href="prob_4.html#T4" target="_self">Th4</a></i>;<br/>
<b>thus </b><a NAME="E6:48_2_5_2_2_3"/>
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:48_2_5_2_2"/><b>then </b><i><font color="Green">E153</font></i>: 
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">DSeq</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E6:48_2_5_2_2"/>
 <a href="prob_1.html#K2">Union</a> <font color="Maroon">DSeq</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sigma</font>
 
<b>by </b><i><a class="ref" href="prob_1.html#T46">PROB_1:46</a></i>;<br/>
<a NAME="E7:48_2_5_2_2"/><b>then </b>
 <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">DSeq</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">Sigma</font>
 
<b>by </b><i><a class="ref" href="card_3.html#D4">CARD_3:def 4</a></i>;<br/>
<b>hence </b><a NAME="E8:48_2_5_2_2"/>
<span class="p1">(<span class="default"><a href="prob_1.html#K2">Union</a> <font color="Maroon">A1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">BSeq</font></span>)</span></span>)</span> is    <a href="prob_4.html#M1" target="_self">thin</a> of <font color="Maroon">P</font>
 <b>by </b><i>, , <a class="txt" href="prob_4.html#E1:48"><i><font color="Green">E24</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">C</font> being    <a href="prob_4.html#M1" target="_self">thin</a> of <font color="Maroon">P</font><b> such that </b><br/><a NAME="E4:48_2_5_2"/><i><font color="Green">E154</font></i>: 
 <a href="prob_1.html#K2">Union</a> <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">BSeq</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font>
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">C</font>
;<br/>

<b>thus </b><a NAME="E5:48_2_5_2"/>
 <a href="prob_1.html#K2">Union</a> <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">BSeq</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>take </b>
 <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">BSeq</font></span>)</span>
;<br/>

<a NAME="E3:48_2_5"/>
 <a href="prob_1.html#K2">Union</a> <font color="Maroon">BSeq</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sigma</font>
 
<b>by </b><i><a class="ref" href="prob_1.html#T46">PROB_1:46</a></i>;<br/>
<b>hence </b><a NAME="E4:48_2_5"/>
(  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">BSeq</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">Sigma</font> &amp;  ex <font color="Olive">C</font> being   <a href="prob_4.html#M1" target="_self">thin</a> of <font color="Maroon">P</font> st  <a href="prob_1.html#K2">Union</a> <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">BSeq</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Olive">C</font> )
 <b>by </b><i><a class="ref" href="card_3.html#D4">CARD_3:def 4</a>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E18:48_2"/>
 <a href="prob_1.html#K2">Union</a> <font color="Maroon">A1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="prob_4.html#T2" target="_self">Th2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:48"/>
<font color="Maroon">D</font> is   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">Omega</font>
 <b>by </b><i>, </i>;<br/>


</div>
