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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="afinsq_1.html#NM1">XFinSequence</a>;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="real_1.html#NM1">Real</a>] means for <font color="Olive">b<sub>1</sub></font> being  <a href="afinsq_1.html#NM1">XFinSequence</a> st  <a href="afinsq_1.html#K1" target="_self">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>];<br/>
<b>consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E2:70_1"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] )
 <b>from </b><i><a class="ref" href="real_1.html#S1">REAL_1:sch 1</a>();<br/></i>
<a NAME="E3:70_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E1:70_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:70_1_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="afinsq_1.html#NM1">XFinSequence</a> st  <a href="afinsq_1.html#K1" target="_self">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>by </b><i><a class="txt" href="afinsq_1.html#E1:70"><i><font color="Green">E31</font></i></a>, <a class="ref" href="afinsq_1.html#T18" target="_self">Th18</a></i>;<br/>
<b>hence </b><a NAME="E2:70_1_1_1"/>
0 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="afinsq_1.html#E2:70_1"><i><font color="Green">E33</font></i></a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:70_1_1_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="afinsq_1.html#NM1">XFinSequence</a>;<br/><b>assume </b><a NAME="E1:70_1_1_2_1"/><i><font color="Green">E36</font></i>: 
 <a href="afinsq_1.html#K1" target="_self">len</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E2:70_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></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="afinsq_1.html#T18" target="_self">Th18</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="afinsq_1.html#NM1">XFinSequence</a>, <font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:70_1_1_2_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K1">^</a> <span class="p1"><a href="afinsq_1.html#K6" target="_self">&lt;%</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="afinsq_1.html#K6" target="_self">%&gt;</a></span>
 <b>by </b><i><a class="ref" href="afinsq_1.html#T44" target="_self">Th44</a></i>;<br/> <a href="afinsq_1.html#K1" target="_self">len</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><a href="afinsq_1.html#K1" target="_self">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" target="_self">len</a> <span class="p2"><a href="afinsq_1.html#K6" target="_self">&lt;%</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="afinsq_1.html#K6" target="_self">%&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="afinsq_1.html#E4:70_1_1_2_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="afinsq_1.html#T20" target="_self">Th20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="afinsq_1.html#K1" target="_self">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="afinsq_1.html#T37" target="_self">Th37</a></i>
;<br/><a NAME="E6:70_1_1_2_1"/><b>then </b>
 <a href="afinsq_1.html#K1" target="_self">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="txt" href="afinsq_1.html#E1:70_1_1_2_1"><i><font color="Green">E36</font></i></a></i>;<br/><a NAME="E7:70_1_1_2_1"/><b>then </b>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font>]
 <b>by </b><i><a class="txt" href="afinsq_1.html#E2:70_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="afinsq_1.html#E1:70_1_1_2"><i><font color="Green">E35</font></i></a></i>;<br/><b>hence </b><a NAME="E8:70_1_1_2_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>]
 <b>by </b><i><a class="txt" href="afinsq_1.html#E2:70"><i><font color="Green">E32</font></i></a>, <a class="txt" href="afinsq_1.html#E4:70_1_1_2_1"><i><font color="Green">E37</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:70_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="afinsq_1.html#E2:70_1"><i><font color="Green">E33</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:70_1_1"/><b>then </b><i><font color="Green">E35</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
;<br/>
<b>thus </b><a NAME="E4:70_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="afinsq_1.html#E1:70_1_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="afinsq_1.html#E3:70_1_1"><i><font color="Green">E35</font></i></a>);</i><br/>


</div><b>end;</b></div>
<a NAME="E4:70_1"/><b>then </b>
 <a href="afinsq_1.html#K1" target="_self">len</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
;<br/>
<b>hence </b><a NAME="E5:70_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font>]
 <b>by </b><i><a class="txt" href="afinsq_1.html#E2:70_1"><i><font color="Green">E33</font></i></a></i>;<br/>


</div>
