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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">c<sub>2</sub></font>, <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="rearran1.html#NM1">RearrangmentGen</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>









<b>assume </b><a NAME="E1:49"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> is <a href="partfun1.html#V1">total</a> &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>5</sub></font>;<br/>
<a NAME="E2:49"/><i><font color="Green">E37</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></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">c<sub>5</sub></font></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <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> <span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <span class="p2">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <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> <span class="p2">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <span class="p3">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:49"/><i><font color="Green">E38</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <span class="p2">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="rearran1.html#E1:49"><i><font color="Green">E36</font></i></a>, <a class="ref" href="rearran1.html#T1" target="_self">Th1</a>, <a class="ref" href="rearran1.html#T36" target="_self">Th36</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</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>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> implies <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <span class="p2">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="rfunct_3.html#K22">FinS</a> <span class="p1">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> );<br/>
<a NAME="E4:49"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E5:49"/><i><font color="Green">E40</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>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:49_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>8</sub></font>]
 ;<br/>

<b>set </b><font color="Maroon">c<sub>9</sub></font> =  <a href="rfunct_3.html#K22">FinS</a> <span class="p1">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>;<br/>
<b>assume </b><a NAME="E2:49_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 ;<br/>

<a NAME="E3:49_1"/><b>then </b><i><font color="Green">E43</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E4:49_1"/><b>then </b><i><font color="Green">E44</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1 &amp; <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E5:49_1"/><b>then </b><i><font color="Green">E45</font></i>: 
( 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <span class="p3">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <span class="p4">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="rearran1.html#E3:49_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E6:49_1"/><i><font color="Green">E46</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <span class="p3">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="rearran1.html#E3:49"><i><font color="Green">E38</font></i></a>, <a class="txt" href="rearran1.html#E3:49_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E7:49_1"/><i><font color="Green">E47</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="rearran1.html#E1:49"><i><font color="Green">E36</font></i></a>, <a class="ref" href="rearran1.html#T21" target="_self">Th21</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_1_1_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_1"><b>case </b></a><a NAME="E1:49_1_1_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E3:49_1_1_1_1"/><i><font color="Green">E49</font></i>: 
<span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="rearran1.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E4:49_1_1_1_1"/>
( <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="rearran1.html#E1:49"><i><font color="Green">E36</font></i></a>, <a class="txt" href="rearran1.html#E5:49_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="rearran1.html#E14"><i><font color="Green">Lemma14</font></i></a>, <a class="ref" href="rearran1.html#T21" target="_self">Th21</a></i>;<br/><a NAME="E5:49_1_1_1_1"/><b>then </b><i><font color="Green">E50</font></i>: 
 <a href="rfunct_3.html#K22">FinS</a> <span class="p1">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="rearran1.html#E1:49_1_1_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="rearran1.html#E3:49_1_1_1_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="rfunct_3.html#T72">RFUNCT_3:72</a></i>;<br/><a NAME="E6:49_1_1_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i><a class="txt" href="rearran1.html#E3:49_1_1_1_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E7:49_1_1_1_1"/><i><font color="Green">E52</font></i>: 
( 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1 &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <span class="p2">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="rearran1.html#E1:49"><i><font color="Green">E36</font></i></a>, <a class="ref" href="rearran1.html#T36" target="_self">Th36</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E8:49_1_1_1_1"/><b>then </b><i><font color="Green">E53</font></i>: 
( 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <span class="p2">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <span class="p3">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="txt" href="rearran1.html#E1:49_1_1_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <span class="p3">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> 1 = 
<span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <span class="p2">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> 1
<b>by </b><i><a class="txt" href="rearran1.html#E1:49_1_1_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="rearran1.html#E7:49_1_1_1_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="rfinseq.html#T19">RFINSEQ:19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> 1
<b>by </b><i><a class="txt" href="rearran1.html#E1:49"><i><font color="Green">E36</font></i></a>, <a class="ref" href="rearran1.html#T25" target="_self">Th25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="rearran1.html#E1:49"><i><font color="Green">E36</font></i></a>, <a class="txt" href="rearran1.html#E6:49_1_1_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="rearran1.html#T22" target="_self">Th22</a></i>
;<br/><b>hence </b><a NAME="E10:49_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <span class="p2">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="rfunct_3.html#K22">FinS</a> <span class="p1">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E5:49_1_1_1_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="rearran1.html#E8:49_1_1_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_2"><b>case </b></a><a NAME="E1:49_1_1_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:49_1_1_1_2"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/><a NAME="E3:49_1_1_1_2"/><b>then </b><i><font color="Green">E48</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E5:49_1_1_1_2"/><i><font color="Green">E49</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span> &amp; <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="rearran1.html#E4:49_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="rearran1.html#T10" target="_self">Th10</a></i>;<br/><a NAME="E6:49_1_1_1_2"/><i><font color="Green">E50</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E3:49_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/><i><font color="Green">E51</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <span class="p3">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <span class="p3">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <span class="p3">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <span class="p2">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T100">RELAT_1:100</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <span class="p2">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rearran1.html#E6:49_1_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <span class="p2">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>
;<br/><a NAME="E8:49_1_1_1_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> = 
<span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="rearran1.html#E1:49"><i><font color="Green">E36</font></i></a>, <a class="txt" href="rearran1.html#E4:49_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="rearran1.html#E3:49_1_1_1_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="rearran1.html#E5:49_1_1_1_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="rearran1.html#T22" target="_self">Th22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <span class="p2">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="rearran1.html#E1:49"><i><font color="Green">E36</font></i></a>, <a class="ref" href="rearran1.html#T25" target="_self">Th25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <span class="p3">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="rearran1.html#E2:49"><i><font color="Green">E37</font></i></a>, <a class="txt" href="rearran1.html#E3:49"><i><font color="Green">E38</font></i></a>, <a class="txt" href="rearran1.html#E2:49_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="rearran1.html#E5:49_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="rfinseq.html#T19">RFINSEQ:19</a></i>
;<br/><a NAME="E10:49_1_1_1_2"/><b>then </b><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <span class="p2">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <span class="p3">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></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"><span class="p3">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="rearran1.html#E6:49_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="rearran1.html#E7:49_1_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="rfinseq.html#T20">RFINSEQ:20</a></i>;<br/><a NAME="E11:49_1_1_1_2"/><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="rearran1.html#E2:49_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="rearran1.html#E14"><i><font color="Green">Lemma14</font></i></a></i>;<br/><a NAME="E12:49_1_1_1_2"/><b>then </b><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/><a NAME="E13:49_1_1_1_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E5:49_1_1_1_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="rearran1.html#E8:49_1_1_1_2"><i><font color="Green">E52</font></i></a></i>;<br/><a NAME="E14:49_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E7:49_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E15:49_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/><a NAME="E16:49_1_1_1_2"/><b>then </b><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <span class="p2">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 <b>by </b><i><a class="txt" href="rearran1.html#E1:49_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="rearran1.html#E4:49_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="rearran1.html#E3:49_1_1_1_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="rearran1.html#E5:49_1_1_1_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="rearran1.html#E10:49_1_1_1_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="rearran1.html#E12:49_1_1_1_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a>, <a class="ref" href="rfunct_3.html#T68">RFUNCT_3:68</a></i>;<br/> <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="rearran1.html#E7:49_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="rearran1.html#E11:49_1_1_1_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/><a NAME="E18:49_1_1_1_2"/><b>then </b>
 <a href="rfunct_3.html#K22">FinS</a> <span class="p1">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 <b>by </b><i><a class="ref" href="rfunct_3.html#D14">RFUNCT_3:def 14</a></i>;<br/><a NAME="E19:49_1_1_1_2"/><b>then </b><i><font color="Green">E57</font></i>: 
 <a href="rfunct_3.html#K22">FinS</a> <span class="p1">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <span class="p2">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 <b>by </b><i><a class="txt" href="rearran1.html#E16:49_1_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="rfinseq.html#T2">RFINSEQ:2</a></i>;<br/><a NAME="E20:49_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <span class="p2">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is <a href="rfinseq.html#V1">non-increasing</a>
 <b>by </b><i><a class="ref" href="rfinseq.html#T33">RFINSEQ:33</a></i>;<br/><b>hence </b><a NAME="E21:49_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <span class="p2">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="rfunct_3.html#K22">FinS</a> <span class="p1">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E19:49_1_1_1_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="rfinseq.html#T36">RFINSEQ:36</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E9:49_1"/>
<span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <span class="p2">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="rfunct_3.html#K22">FinS</a> <span class="p1">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:49"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</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="rearran1.html#E4:49"><i><font color="Green">E39</font></i></a>, <a class="txt" href="rearran1.html#E5:49"><i><font color="Green">E40</font></i></a>);<br/></i>
<b>hence </b><a NAME="E7:49"/>
<span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <span class="p2">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="rfunct_3.html#K22">FinS</a> <span class="p1">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E1:49"><i><font color="Green">E36</font></i></a>, <a class="txt" href="rearran1.html#E2:49"><i><font color="Green">E37</font></i></a>, <a class="txt" href="rearran1.html#E3:49"><i><font color="Green">E38</font></i></a></i>;<br/>


</div>
