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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>( ( 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> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Olive">a</font> is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  ) &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="modelc_1.html#K7" target="_self">atom.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> ) &amp; ( for <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> holds <br/> <a href="modelc_1.html#K8" target="_self">'not'</a> <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Olive">p</font> <a href="modelc_1.html#K9" target="_self">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> ) &amp; ( for <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> holds <br/> <a href="modelc_1.html#K10" target="_self">EX</a> <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> ) &amp; ( for <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> holds <br/> <a href="modelc_1.html#K11" target="_self">EG</a> <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Olive">p</font> <a href="modelc_1.html#K12" target="_self">EU</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> ) );<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">D</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">D</font>] holds <br/><font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">D</font>;<br/>
<b>consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:12_1_1"/><i><font color="Green">E27</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">Y</font> iff ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K5">NAT</a>  <a href="finseq_1.html#K13">*</a>  &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">a</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_1"><b>now </b></a><div class="add"><b>set </b><font color="Maroon">a</font> =  <a href="modelc_1.html#K7" target="_self">atom.</a> 0;<br/><a NAME="E1:12_1_1_1"/><i><font color="Green">E28</font></i>: 
 <a href="modelc_1.html#K7" target="_self">atom.</a> 0 <a href="hidden.html#R2">in</a> <a href="numbers.html#K5">NAT</a>  <a href="finseq_1.html#K13">*</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/><a NAME="E2:12_1_1_1"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">D</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">D</font>] holds <br/> <a href="modelc_1.html#K7" target="_self">atom.</a> 0 <a href="hidden.html#R2">in</a> <font color="Olive">D</font>
 ;<br/><b>take </b><font color="Maroon">b</font> =  <a href="modelc_1.html#K7" target="_self">atom.</a> 0;<br/><b>thus </b><a NAME="E3:12_1_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">Y</font> = <font color="Maroon">Y</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>take </b>
<font color="Maroon">Y</font>
;<br/>

<b>thus </b><a NAME="E5:12_1_1"/>
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> <font color="Maroon">Y</font> holds <br/><font color="Olive">a</font> is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_2"><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="E1:12_1_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<a NAME="E2:12_1_1_2"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K5">NAT</a>  <a href="finseq_1.html#K13">*</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:12_1_1_2"/>
<font color="Maroon">a</font> is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>


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

<b>thus </b><a NAME="E6:12_1_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="modelc_1.html#K7" target="_self">atom.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_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="E1:12_1_1_3"/><i><font color="Green">E52</font></i>: 
 <a href="modelc_1.html#K7" target="_self">atom.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K5">NAT</a>  <a href="finseq_1.html#K13">*</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E2:12_1_1_3"/>
for <font color="Olive">D</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">D</font>] holds <br/> <a href="modelc_1.html#K7" target="_self">atom.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Olive">D</font>
 
;<br/>
<b>hence </b><a NAME="E3:12_1_1_3"/>
 <a href="modelc_1.html#K7" target="_self">atom.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i>, </i>;<br/>


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

<b>thus </b><a NAME="E7:12_1_1"/>
for <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/> <a href="modelc_1.html#K8" target="_self">'not'</a> <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:12_1_1_4"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<a NAME="E2:12_1_1_4"/><i><font color="Green">E54</font></i>: 
 <a href="modelc_1.html#K8" target="_self">'not'</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K5">NAT</a>  <a href="finseq_1.html#K13">*</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E3:12_1_1_4"/>
for <font color="Olive">D</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">D</font>] holds <br/> <a href="modelc_1.html#K8" target="_self">'not'</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_4_1"><b>proof </b></a><div class="add">

<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>assume </b><a NAME="E1:12_1_1_4_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">D</font>]
 ;<br/>

<a NAME="E2:12_1_1_4_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E3:12_1_1_4_1"/>
 <a href="modelc_1.html#K8" target="_self">'not'</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:12_1_1_4"/>
 <a href="modelc_1.html#K8" target="_self">'not'</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i>, </i>;<br/>


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

<b>thus </b><a NAME="E8:12_1_1"/>
for <font color="Olive">q</font>, <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/><font color="Olive">q</font> <a href="modelc_1.html#K9" target="_self">'&amp;'</a> <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">p</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:12_1_1_5"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 ;<br/>

<a NAME="E2:12_1_1_5"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">q</font> <a href="modelc_1.html#K9" target="_self">'&amp;'</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K5">NAT</a>  <a href="finseq_1.html#K13">*</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E3:12_1_1_5"/>
for <font color="Olive">D</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">D</font>] holds <br/><font color="Maroon">q</font> <a href="modelc_1.html#K9" target="_self">'&amp;'</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_5_1"><b>proof </b></a><div class="add">

<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>assume </b><a NAME="E1:12_1_1_5_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">D</font>]
 ;<br/>

<a NAME="E2:12_1_1_5_1"/><b>then </b>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E3:12_1_1_5_1"/>
<font color="Maroon">q</font> <a href="modelc_1.html#K9" target="_self">'&amp;'</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:12_1_1_5"/>
<font color="Maroon">q</font> <a href="modelc_1.html#K9" target="_self">'&amp;'</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i>, </i>;<br/>


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

<b>thus </b><a NAME="E9:12_1_1"/>
for <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/> <a href="modelc_1.html#K10" target="_self">EX</a> <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:12_1_1_6"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<a NAME="E2:12_1_1_6"/><i><font color="Green">E60</font></i>: 
 <a href="modelc_1.html#K10" target="_self">EX</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K5">NAT</a>  <a href="finseq_1.html#K13">*</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E3:12_1_1_6"/>
for <font color="Olive">D</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">D</font>] holds <br/> <a href="modelc_1.html#K10" target="_self">EX</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_6_1"><b>proof </b></a><div class="add">

<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>assume </b><a NAME="E1:12_1_1_6_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">D</font>]
 ;<br/>

<a NAME="E2:12_1_1_6_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E3:12_1_1_6_1"/>
 <a href="modelc_1.html#K10" target="_self">EX</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:12_1_1_6"/>
 <a href="modelc_1.html#K10" target="_self">EX</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i>, </i>;<br/>


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

<b>thus </b><a NAME="E10:12_1_1"/>
for <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/> <a href="modelc_1.html#K11" target="_self">EG</a> <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:12_1_1_7"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<a NAME="E2:12_1_1_7"/><i><font color="Green">E63</font></i>: 
 <a href="modelc_1.html#K11" target="_self">EG</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K5">NAT</a>  <a href="finseq_1.html#K13">*</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E3:12_1_1_7"/>
for <font color="Olive">D</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">D</font>] holds <br/> <a href="modelc_1.html#K11" target="_self">EG</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_7_1"><b>proof </b></a><div class="add">

<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>assume </b><a NAME="E1:12_1_1_7_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">D</font>]
 ;<br/>

<a NAME="E2:12_1_1_7_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E3:12_1_1_7_1"/>
 <a href="modelc_1.html#K11" target="_self">EG</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:12_1_1_7"/>
 <a href="modelc_1.html#K11" target="_self">EG</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i>, </i>;<br/>


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

<b>thus </b><a NAME="E11:12_1_1"/>
for <font color="Olive">q</font>, <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/><font color="Olive">q</font> <a href="modelc_1.html#K12" target="_self">EU</a> <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">p</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:12_1_1_8"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 ;<br/>

<a NAME="E2:12_1_1_8"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">q</font> <a href="modelc_1.html#K12" target="_self">EU</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K5">NAT</a>  <a href="finseq_1.html#K13">*</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E3:12_1_1_8"/>
for <font color="Olive">D</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">D</font>] holds <br/><font color="Maroon">q</font> <a href="modelc_1.html#K12" target="_self">EU</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_8_1"><b>proof </b></a><div class="add">

<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>assume </b><a NAME="E1:12_1_1_8_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">D</font>]
 ;<br/>

<a NAME="E2:12_1_1_8_1"/><b>then </b>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E3:12_1_1_8_1"/>
<font color="Maroon">q</font> <a href="modelc_1.html#K12" target="_self">EU</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:12_1_1_8"/>
<font color="Maroon">q</font> <a href="modelc_1.html#K12" target="_self">EU</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i>, </i>;<br/>


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

<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>assume </b><a NAME="E12:12_1_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">D</font>]
 ;<br/>

<b>let </b><font color="Maroon">a</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="E13:12_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<b>hence </b><a NAME="E14:12_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>by </b><i>, </i>;<br/>


</div>
