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<b>defpred </b><font color="Maroon">S<sub>3</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">g1</font>, <font color="Olive">g2</font> being <a href="rfinseq.html#V1" target="_self">non-increasing</a>  <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>   st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">g1</font> &amp; <font color="Olive">g1</font>,<font color="Olive">g2</font> <a href="rfinseq.html#R1" target="_self">are_fiberwise_equipotent</a>  holds <br/><font color="Olive">g1</font> <a href="hidden.html#R1">=</a> <font color="Olive">g2</font>;<br/>
<a NAME="E1:44"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">g1</font> be  <a href="rfinseq.html#V1" target="_self">non-increasing</a>  <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">g2</font> be  <a href="rfinseq.html#V1" target="_self">non-increasing</a>  <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:44_1"/><i><font color="Green">E32</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> <a href="hidden.html#R1">=</a> 0
 <b>and </b><br/><a NAME="E2:44_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">g1</font>,<font color="Maroon">g2</font> <a href="rfinseq.html#R1" target="_self">are_fiberwise_equipotent</a> 
 ;<br/>

<a NAME="E3:44_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font>
 
<b>by </b><i><a class="ref" href="rfinseq.html#T1" target="_self">Th1</a>, </i>;<br/>
<a NAME="E4:44_1"/><b>then </b>
( <font color="Maroon">g1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> <a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">g2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> <a href="numbers.html#K1">REAL</a>  )
 
<b>by </b><i><a class="txt" href="rfinseq.html#E1:44_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="finseq_1.html#T32">FINSEQ_1:32</a></i>;<br/>
<b>hence </b><a NAME="E5:44_1"/>
<font color="Maroon">g1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:44"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:44_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font>]
 ;<br/>

<b>let </b><font color="Maroon">g1</font> be  <a href="rfinseq.html#V1" target="_self">non-increasing</a>  <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">g2</font> be  <a href="rfinseq.html#V1" target="_self">non-increasing</a>  <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:44_2"/><i><font color="Green">E37</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>and </b><br/><a NAME="E3:44_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">g1</font>,<font color="Maroon">g2</font> <a href="rfinseq.html#R1" target="_self">are_fiberwise_equipotent</a> 
 ;<br/>

<a NAME="E4:44_2"/><i><font color="Green">E39</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font>
 
<b>by </b><i><a class="txt" href="rfinseq.html#E2:44"><i><font color="Green">E35</font></i></a>, </i>;<br/>
<a NAME="E5:44_2"/><b>then </b><i><font color="Green">E40</font></i>: 
( <font color="Maroon">g1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font></span>)</span> &amp; <font color="Maroon">g1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font>,<font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font> <a href="rfinseq.html#R1" target="_self">are_fiberwise_equipotent</a>  )
 
<b>by </b><i>, <a class="txt" href="rfinseq.html#E2:44"><i><font color="Green">E35</font></i></a>, </i>;<br/>
<a NAME="E6:44_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E7:44_2"/><b>then </b><i><font color="Green">E41</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="txt" href="rfinseq.html#E1:44_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g1n</font> = <font color="Maroon">g1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font>, <font color="Maroon">g2n</font> = <font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font> as  <a href="rfinseq.html#V1" target="_self">non-increasing</a>  <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">r</font> = <font color="Maroon">g1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/>
<a NAME="E9:44_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">g1n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2n</font>
 
<b>by </b><i><a class="ref" href="rfinseq.html#T1" target="_self">Th1</a>, <a class="txt" href="rfinseq.html#E2:44_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="rfinseq.html#E3:44_2"><i><font color="Green">E38</font></i></a></i>;<br/>
<a NAME="E10:44_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g1</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g1</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> )
 
<b>by </b><i>, <a class="txt" href="rfinseq.html#E1:44_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="rfinseq.html#E2:44_2"><i><font color="Green">E37</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E11:44_2"/>
<font color="Maroon">g1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font>
 <b>by </b><i><a class="txt" href="rfinseq.html#E4:44_2"><i><font color="Green">E39</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:44"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="rfinseq.html#E1:44"><i><font color="Green">E31</font></i></a>, );</i><br/>


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