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

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/>



<b>assume </b><a NAME="E1:44"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#R2">just_once_values</a> <font color="Maroon">x</font>
 ;<br/>

<a NAME="E2:44"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="finseq_4.html#T7">FINSEQ_4:7</a></i>;<br/>
<a NAME="E3:44"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K3">Rev</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T60">FINSEQ_5:60</a></i>;<br/>
<a NAME="E4:44"/><i><font color="Green">E33</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="finseq_4.html#K5">..</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K3">Rev</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:44"/><i><font color="Green">E34</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K3">Rev</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K6">-|</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K6">-|</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_5.html#D3">FINSEQ_5:def 3</a></i>;<br/>
<i><font color="Green">E36</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K6">-|</a> <font color="Maroon">x</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
<b>by </b><i>, <a class="ref" href="finseq_4.html#T46">FINSEQ_4:46</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="finseq_4.html#K5">..</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K3">Rev</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K3">Rev</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="finseq_4.html#K5">..</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K3">Rev</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_5.html#D3">FINSEQ_5:def 3</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</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">E37</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K3">Rev</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K6">-|</a> <font color="Maroon">x</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:44_2"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K6">-|</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_5.html#T60">FINSEQ_5:60</a></i>;<br/><b>consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:44_2"/>
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>and </b><br/><a NAME="E5:44_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K6">-|</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D6">FINSEQ_4:def 6</a></i>;<br/><a NAME="E6:44_2"/><i><font color="Green">E41</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K6">-|</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i>, <a class="ref" href="finseq_4.html#T46">FINSEQ_4:46</a></i>;<br/><a NAME="E7:44_2"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="finseq_6.html#T7" target="_self">Th7</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E8:44_2"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E9:44_2"/><b>then </b><i><font color="Green">E49</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E10:44_2"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E11:44_2"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="real_2.html#T105">REAL_2:105</a></i>;<br/><a NAME="E12:44_2"/><b>then </b><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E13:44_2"/><i><font color="Green">E51</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="finseq_6.html#T6" target="_self">Th6</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E14:44_2"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="finseq_4.html#K5">..</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K3">Rev</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E15:44_2"/><b>then </b><i><font color="Green">E52</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="finseq_4.html#K5">..</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K3">Rev</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="finseq_6.html#E15:44_2"><i><font color="Green">E52</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E16:44_2"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="txt" href="finseq_6.html#E15:44_2"><i><font color="Green">E52</font></i></a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E17:44_2"/><b>then </b><i><font color="Green">E62</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_6.html#T8" target="_self">Th8</a>, <a class="ref" href="finseq_6.html#T9" target="_self">Th9</a>, <a class="txt" href="finseq_6.html#E12:44_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="finseq_6.html#E13:44_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E18:44_2"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="finseq_6.html#E9:44_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E19:44_2"/><b>then </b>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="finseq_4.html#K5">..</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K3">Rev</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="finseq_6.html#E8:44_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E20:44_2"/><b>then </b>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="finseq_4.html#K5">..</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K3">Rev</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E21:44_2"/><b>then </b>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="finseq_4.html#K5">..</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K3">Rev</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K3">Rev</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_5.html#D3">FINSEQ_5:def 3</a></i>;<br/><a NAME="E22:44_2"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="finseq_4.html#K5">..</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K3">Rev</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K3">Rev</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_6.html#T10" target="_self">Th10</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="finseq_5.html#K3">Rev</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K6">-|</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K6">-|</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_6.html#T6" target="_self">Th6</a>, <a class="ref" href="finseq_6.html#T9" target="_self">Th9</a>, <a class="ref" href="finseq_5.html#D3">FINSEQ_5:def 3</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="finseq_4.html#K5">..</a> <span class="p5">(<span class="default"><a href="finseq_5.html#K3">Rev</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="finseq_6.html#E8:44_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_6.html#T8" target="_self">Th8</a>, , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_5.html#K3">Rev</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="finseq_4.html#K5">..</a> <span class="p3">(<span class="default"><a href="finseq_5.html#K3">Rev</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_6.html#T11" target="_self">Th11</a>, <a class="ref" href="finseq_5.html#D3">FINSEQ_5:def 3</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:44"/>
 <a href="finseq_5.html#K3">Rev</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K6">-|</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K3">Rev</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_4.html#K7">|--</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="finseq_6.html#T4" target="_self">Th4</a>, <a class="ref" href="finseq_6.html#T5" target="_self">Th5</a>, <a class="ref" href="finseq_4.html#D7">FINSEQ_4:def 7</a></i>;<br/>


</div>
