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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">x1</font>, <font color="Olive">x2</font>, <font color="Olive">y1</font>, <font color="Olive">y2</font> being  <a href="finseq_1.html#NM1">FinSequence</a>  st  <a href="finseq_1.html#K3">len</a> <font color="Olive">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R1">=</a> <font color="Olive">x1</font> <a href="finseq_1.html#K7">^</a> <font color="Olive">x2</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1">=</a> <font color="Olive">y1</font> <a href="finseq_1.html#K7">^</a> <font color="Olive">y2</font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">y1</font> <a href="finseq_1.html#K7">^</a> <font color="Olive">x2</font>];<br/>
<a NAME="E1:52_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be   <a href="finseq_1.html#NM1">FinSequence</a>, <font color="Maroon">x2</font> be   <a href="finseq_1.html#NM1">FinSequence</a>, <font color="Maroon">y1</font> be   <a href="finseq_1.html#NM1">FinSequence</a>, <font color="Maroon">y2</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/>

<b>assume </b><a NAME="E1:52_1_1"/><i><font color="Green">E58</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">x2</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">y2</font> )
 ;<br/>

<a NAME="E2:52_1_1"/><b>then </b>
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<b>hence </b><a NAME="E3:52_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">y1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">x2</font>]
 <b>by </b><i>, <a class="txt" href="pua2mss1.html#E2:52"><i><font color="Green">E36</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:52_1"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:52_1_2"/><i><font color="Green">E72</font></i>: 
for <font color="Olive">x1</font>, <font color="Olive">x2</font>, <font color="Olive">y1</font>, <font color="Olive">y2</font> being  <a href="finseq_1.html#NM1">FinSequence</a>  st  <a href="finseq_1.html#K3">len</a> <font color="Olive">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R1">=</a> <font color="Olive">x1</font> <a href="finseq_1.html#K7">^</a> <font color="Olive">x2</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1">=</a> <font color="Olive">y1</font> <a href="finseq_1.html#K7">^</a> <font color="Olive">y2</font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">y1</font> <a href="finseq_1.html#K7">^</a> <font color="Olive">x2</font>]
 ;<br/>

<b>let </b><font color="Maroon">x1</font> be   <a href="finseq_1.html#NM1">FinSequence</a>, <font color="Maroon">x2</font> be   <a href="finseq_1.html#NM1">FinSequence</a>, <font color="Maroon">y1</font> be   <a href="finseq_1.html#NM1">FinSequence</a>, <font color="Maroon">y2</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/>

<b>assume </b><a NAME="E2:52_1_2"/><i><font color="Green">E73</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">x2</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">y2</font> )
 ;<br/>

<a NAME="E3:52_1_2"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">y1</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="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<b>then consider </b><font color="Maroon">x'</font> being   <a href="finseq_1.html#NM1">FinSequence</a>, <font color="Maroon">xx</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:52_1_2"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">xx</font></span><a href="finseq_1.html#K9">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T63">FINSEQ_1:63</a></i>;<br/>
<b>consider </b><font color="Maroon">y'</font> being   <a href="finseq_1.html#NM1">FinSequence</a>, <font color="Maroon">yy</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:52_1_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">yy</font></span><a href="finseq_1.html#K9">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="pua2mss1.html#T1" target="_self">Th1</a>, <a class="ref" href="finseq_1.html#T63">FINSEQ_1:63</a></i>;<br/>
<a NAME="E8:52_1_2"/><i><font color="Green">E126</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">x1</font></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">y1</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E9:52_1_2"/>
(  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">xx</font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">yy</font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>
<a NAME="E10:52_1_2"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">x'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">y'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E11:52_1_2"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">xx</font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">x2</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">yy</font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">y2</font></span>)</span> &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x1</font> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x1</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">x'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">xx</font> &amp; <font color="Maroon">y1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">y'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">yy</font> )
 
<b>by </b><i>, , , , <a class="ref" href="finseq_1.html#T6">FINSEQ_1:6</a>, <a class="ref" href="finseq_1.html#T39">FINSEQ_1:39</a>, <a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a>, <a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>;<br/>
<a NAME="E12:52_1_2"/><b>then </b>
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">y'</font> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">xx</font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">x2</font></span>)</span>] &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">xx</font> &amp; <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">yy</font> )
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E3:52"><i><font color="Green">E37</font></i></a>, , , <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/>
<a NAME="E13:52_1_2"/><b>then </b>
( <font color="Maroon">P<sub>1</sub></font>[<span class="p1">(<span class="default"><font color="Maroon">y'</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">xx</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">x2</font>] &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon">xx</font>,<font color="Maroon">yy</font>] )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/>
<b>hence </b><a NAME="E14:52_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">y1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">x2</font>]
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:52_1"/><i><font color="Green">E127</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, <a class="txt" href="pua2mss1.html#E2:52"><i><font color="Green">E36</font></i></a>);<br/></i>
<a NAME="E4:52_1"/>
( <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() <a href="finseq_1.html#K7">^</a> <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() <a href="finseq_1.html#K7">^</a> <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/>
<b>hence </b><a NAME="E5:52_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>2</sub></font>()]
 <b>by </b><i>, <a class="txt" href="pua2mss1.html#E3:52"><i><font color="Green">E37</font></i></a></i>;<br/>


</div>
