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<b>let </b><font color="Maroon" title="c1">p</font> be   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>;<br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>for <font color="Olive" title="b1">p</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>];<br/>
<b>consider </b><font color="Maroon" title="c2">X</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E2:80_1"/><i><font color="Green" title="E49">A3</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> iff <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>] )
 <b>from </b><i><a class="ref" href="real_1.html#S1" title="REAL_1:sch.1">REAL_1:sch 1</a>();<br/></i>
<a NAME="E3:80_1"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_1"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>$1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>;<br/>
<a NAME="E1:80_1_1"/><i><font color="Green" title="E50">A4</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:80_1_1_1"/>
for <font color="Olive" title="b1">q</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">q</font>]
 
<b>by </b><i><a class="txt" href="finseq_1.html#E1:80"><i><font color="Green" title="E47">A1</font></i></a>, <a class="txt" href="finseq_1.html#E32"><i><font color="Green" title="E20">Lm4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:80_1_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E2:80_1"><i><font color="Green" title="E49">A3</font></i></a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:80_1_1_2"/><i><font color="Green" title="E51">A5</font></i>: 
<font color="Maroon" title="c3">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c4">n1</font> = <font color="Maroon" title="c3">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">q</font> be   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>;<br/><b>assume </b><a NAME="E1:80_1_1_2_1"/><i><font color="Green" title="E52">A6</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><a NAME="E2:80_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c5">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="finseq_1.html#E32"><i><font color="Green" title="E20">Lm4</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">r</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>, <font color="Maroon" title="c7">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:80_1_1_2_1"/><i><font color="Green" title="E53">A7</font></i>: 
<font color="Maroon" title="c5">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T63" target="_self" title="FINSEQ_1:th.63">Th63</a></i>;<br/> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">q</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="finseq_1.html#E4:80_1_1_2_1"><i><font color="Green" title="E53">A7</font></i></a>, <a class="ref" href="finseq_1.html#T35" target="_self" title="FINSEQ_1:th.35">Th35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T56" target="_self" title="FINSEQ_1:th.56">Th56</a></i>
;<br/><a NAME="E6:80_1_1_2_1"/><b>then </b>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c6">r</font>]
 <b>by </b><i><a class="txt" href="finseq_1.html#E2:80_1"><i><font color="Green" title="E49">A3</font></i></a>, <a class="txt" href="finseq_1.html#E1:80_1_1_2"><i><font color="Green" title="E51">A5</font></i></a>, <a class="txt" href="finseq_1.html#E1:80_1_1_2_1"><i><font color="Green" title="E52">A6</font></i></a></i>;<br/><b>hence </b><a NAME="E7:80_1_1_2_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">q</font>]
 <b>by </b><i><a class="txt" href="finseq_1.html#E2:80"><i><font color="Green" title="E48">A2</font></i></a>, <a class="txt" href="finseq_1.html#E4:80_1_1_2_1"><i><font color="Green" title="E53">A7</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E4:80_1_1_2"/><b>then </b>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ;<br/><b>hence </b><a NAME="E5:80_1_1_2"/>
<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E2:80_1"><i><font color="Green" title="E49">A3</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:80_1_1"/><b>then </b><i><font color="Green" title="E51">A8</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
;<br/>
<b>thus </b><a NAME="E4:80_1_1"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="finseq_1.html#E1:80_1_1"><i><font color="Green" title="E50">A4</font></i></a>, <a class="txt" href="finseq_1.html#E3:80_1_1"><i><font color="Green" title="E51">A8</font></i></a>);</i><br/>


</div><b>end;</b></div>
<a NAME="E4:80_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 
;<br/>
<b>hence </b><a NAME="E5:80_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">p</font>]
 <b>by </b><i><a class="txt" href="finseq_1.html#E2:80_1"><i><font color="Green" title="E49">A3</font></i></a></i>;<br/>


</div>
