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

<b>let </b><font color="Maroon">D</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</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">F</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">D</font>, <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">B</font> be   <a href="rearran1.html#NM1">RearrangmentGen</a> of <font color="Maroon">C</font>;<br/>







<b>assume </b><a NAME="E1:28"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">F</font> is <a href="partfun1.html#V1">total</a> &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">D</font> )
 ;<br/>

<b>set </b><font color="Maroon">fd</font> =  <a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">D</font>;<br/>
<b>set </b><font color="Maroon">p</font> =  <a href="rearran1.html#K3" target="_self">Rland</a> <font color="Maroon">F</font>,<font color="Maroon">B</font>;<br/>
<b>set </b><font color="Maroon">mf</font> =  <a href="rfinseq.html#K2">MIM</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">D</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">h</font> =  <a href="rfunct_3.html#K16">CHI</a> <font color="Maroon">B</font>,<font color="Maroon">C</font>;<br/>
<a NAME="E2:28"/><i><font color="Green">E34</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rfinseq.html#K2">MIM</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">D</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"><a href="rfunct_3.html#K16">CHI</a> <font color="Maroon">B</font>,<font color="Maroon">C</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K16">CHI</a> <font color="Maroon">B</font>,<font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">B</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rfinseq.html#K2">MIM</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">D</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"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">D</font></span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="rfinseq.html#D3">RFINSEQ:def 3</a>, <a class="ref" href="rfunct_3.html#D6">RFUNCT_3:def 6</a></i>;<br/>
<a NAME="E3:28"/><i><font color="Green">E42</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rearran1.html#K3" target="_self">Rland</a> <font color="Maroon">F</font>,<font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> )
 
<b>by </b><i>, , <a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">fd'</font> =  <a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">D</font>, <font color="Maroon">F'</font> = <font color="Maroon">F</font> as  <a href="finset_1.html#V1">finite</a> <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="ref" href="finset_1.html#T29">FINSET_1:29</a></i>;<br/>
<b>reconsider </b><font color="Maroon">dfd</font> =  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fd'</font>, <font color="Maroon">dF</font> =  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F'</font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E6:28"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">F</font> <a href="rfunct_3.html#K23">|</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T97">RELAT_1:97</a></i>;<br/>
<a NAME="E7:28"/><i><font color="Green">E44</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">D</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> <font color="Maroon">F</font>,<font color="Maroon">D</font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">B</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">B</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:28"/>
 <a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">D</font>,<font color="Maroon">F</font> <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="E9:28"/><b>then </b>
(  <a href="card_1.html#K4">card</a> <font color="Maroon">dfd</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">dF</font> &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">dfd</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">D</font></span>)</span> &amp; <font color="Maroon">D</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="finseq_1.html#T78">FINSEQ_1:78</a>, <a class="ref" href="rfinseq.html#T10">RFINSEQ:10</a></i>;<br/>
<a NAME="E10:28"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">D</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="card_2.html#T59">CARD_2:59</a></i>;<br/>
<a NAME="E11:28"/><b>then </b>
0 <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> <font color="Maroon">F</font>,<font color="Maroon">D</font></span>)</span>
 
;<br/>
<a NAME="E12:28"/><b>then </b><i><font color="Green">E45</font></i>: 
0 <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="rfunct_3.html#K22">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">D</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E13:28"/><b>then </b><i><font color="Green">E49</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> <font color="Maroon">F</font>,<font color="Maroon">D</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">B</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>thus </b><a NAME="E14:28"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="rearran1.html#K3" target="_self">Rland</a> <font color="Maroon">F</font>,<font color="Maroon">B</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">D</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="rearran1.html#K3" target="_self">Rland</a> <font color="Maroon">F</font>,<font color="Maroon">B</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E3:28_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rearran1.html#K3" target="_self">Rland</a> <font color="Maroon">F</font>,<font color="Maroon">B</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="rearran1.html#K3" target="_self">Rland</a> <font color="Maroon">F</font>,<font color="Maroon">B</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">B</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E4:28_1"/><i><font color="Green">E52</font></i>: 
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">n</font> =  <a href="finseq_1.html#K3">len</a> <font color="Maroon">B</font>;<br/>

<a NAME="E1:28_1_1"/>
<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:28_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">B</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 <b>by </b><i>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">mi</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:28_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">mi</font>] &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">mi</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>(<a class="txt" href="rearran1.html#E1:28"><i><font color="Green">E33</font></i></a>);<br/></i>
<a NAME="E7:28_1"/><i><font color="Green">E55</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">mi</font> &amp; <font color="Maroon">mi</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">B</font> )
 
<b>by </b><i><a class="txt" href="rearran1.html#E2:28"><i><font color="Green">E34</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E8:28_1"/><b>then </b>
 <a href="xxreal_0.html#K4">max</a> 0,<span class="p1">(<span class="default"><font color="Maroon">mi</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">mi</font> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="ref" href="finseq_2.html#T4">FINSEQ_2:4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">m1</font> = <font color="Maroon">mi</font> <a href="real_1.html#K5">-</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T5">FINSEQ_2:5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_1_2_1"/>
( <font color="Maroon">mi</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">mi</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_2_1_1"><b>case </b></a><a NAME="E1:28_1_2_1_1"/>
<font color="Maroon">mi</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:28_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="rearran1.html#K3" target="_self">Rland</a> <font color="Maroon">F</font>,<font color="Maroon">B</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">D</font></span>)</span> <a href="seq_1.html#K2">.</a> 1
 <b>by </b><i>, <a class="txt" href="rearran1.html#E2:28"><i><font color="Green">E34</font></i></a>, </i>;<br/><b>hence </b><a NAME="E3:28_1_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">D</font></span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E1"><i><font color="Green">Lemma31</font></i></a>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_2_1_2"><b>case </b></a><a NAME="E1:28_1_2_1_2"/>
<font color="Maroon">mi</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:28_1_2_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">mi</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:28_1_2_1_2"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">mi</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:28_1_2_1_2"/><b>then </b><i><font color="Green">E56</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E5:28_1_2_1_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">m1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">mi</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T46">XREAL_1:46</a></i>;<br/><a NAME="E6:28_1_2_1_2"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">m1</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:28_1_2_1_2"/>
<font color="Maroon">m1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">B</font>
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E8:28_1_2_1_2"/><b>then </b>
<font color="Maroon">m1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="rearran1.html#D1" target="_self">Def1</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E9:28_1_2_1_2"/><b>then </b>
not <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E2:28"><i><font color="Green">E34</font></i></a>, </i>;<br/><a NAME="E10:28_1_2_1_2"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">m1</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"><font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font></span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E2:28"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E11:28_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="rearran1.html#K3" target="_self">Rland</a> <font color="Maroon">F</font>,<font color="Maroon">B</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">D</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">m1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="rearran1.html#D1" target="_self">Def1</a>, <a class="ref" href="rearran1.html#D2" target="_self">Def2</a>, </i>;<br/><b>hence </b><a NAME="E12:28_1_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">D</font></span>)</span>
 <b>by </b><i>, , , <a class="txt" href="rearran1.html#E2:28"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E11:28_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">D</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E15:28"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">D</font></span>)</span>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <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="rfunct_3.html#K22">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">D</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">D</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> );<br/>
<a NAME="E16:28"/><i><font color="Green">E60</font></i>: 
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>by </b><i>, <a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/>
<b>consider </b><font color="Maroon">mi</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E18:28"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">mi</font>] &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">mi</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>(<a class="txt" href="rearran1.html#E1:28"><i><font color="Green">E33</font></i></a>);<br/></i>
<a NAME="E19:28"/><i><font color="Green">E63</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">mi</font> &amp; <font color="Maroon">mi</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="rfunct_3.html#K22">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">D</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="rearran1.html#E2:28"><i><font color="Green">E34</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E20:28"/><b>then </b>
 <a href="xxreal_0.html#K4">max</a> 0,<span class="p1">(<span class="default"><font color="Maroon">mi</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">mi</font> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="ref" href="finseq_2.html#T4">FINSEQ_2:4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">m1</font> = <font color="Maroon">mi</font> <a href="real_1.html#K5">-</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T5">FINSEQ_2:5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_2_1"/>
( <font color="Maroon">mi</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">mi</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_1_1"><b>case </b></a><a NAME="E1:28_2_1_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">mi</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:28_2_1_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, , <a class="txt" href="rearran1.html#E1"><i><font color="Green">Lemma31</font></i></a>, </i>;<br/><b>consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> 1;<br/><a NAME="E4:28_2_1_1"/>
<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> 1 <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 <b>by </b><i>, , <a class="txt" href="rearran1.html#E1"><i><font color="Green">Lemma31</font></i></a>, </i>;<br/><b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font> <b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E6:28_2_1_1"/>
<span class="p1">(<span class="default"><a href="rearran1.html#K3" target="_self">Rland</a> <font color="Maroon">F</font>,<font color="Maroon">B</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">D</font></span>)</span> <a href="seq_1.html#K2">.</a> 1
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E7:28_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="rearran1.html#K3" target="_self">Rland</a> <font color="Maroon">F</font>,<font color="Maroon">B</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="rearran1.html#E2:28"><i><font color="Green">E34</font></i></a>, <a class="ref" href="rearran1.html#D1" target="_self">Def1</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_1_2"><b>case </b></a><a NAME="E1:28_2_1_2"/>
<font color="Maroon">mi</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:28_2_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">mi</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:28_2_1_2"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">mi</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:28_2_1_2"/><b>then </b><i><font color="Green">E66</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E5:28_2_1_2"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">m1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">mi</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T46">XREAL_1:46</a></i>;<br/><a NAME="E6:28_2_1_2"/><b>then </b><i><font color="Green">E91</font></i>: 
<font color="Maroon">m1</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="rfunct_3.html#K22">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">D</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:28_2_1_2"/>
( <font color="Maroon">m1</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="rfunct_3.html#K22">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">D</font></span>)</span> &amp; <font color="Maroon">m1</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="rfunct_3.html#K22">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">D</font></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E8:28_2_1_2"/><b>then </b>
( <font color="Maroon">m1</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="rfunct_3.html#K22">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">D</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 &amp; <font color="Maroon">m1</font> <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> <font color="Maroon">F</font>,<font color="Maroon">D</font></span>)</span> )
 <b>by </b><i><a class="ref" href="rearran1.html#D1" target="_self">Def1</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E9:28_2_1_2"/><b>then </b><i><font color="Green">E92</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">m1</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"><font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font></span>)</span> <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="rearran1.html#D1" target="_self">Def1</a>, </i>;<br/><b>consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">m1</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"><font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font></span>)</span>;<br/><a NAME="E11:28_2_1_2"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">mi</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> )
 <b>by </b><i>, , <a class="txt" href="rearran1.html#E2:28"><i><font color="Green">E34</font></i></a>, <a class="txt" href="rearran1.html#E4"><i><font color="Green">Lemma39</font></i></a>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font> ;<br/><a NAME="E13:28_2_1_2"/>
<span class="p1">(<span class="default"><a href="rearran1.html#K3" target="_self">Rland</a> <font color="Maroon">F</font>,<font color="Maroon">B</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">D</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">m1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="rearran1.html#D1" target="_self">Def1</a>, <a class="ref" href="rearran1.html#D2" target="_self">Def2</a>, <a class="txt" href="rearran1.html#E4"><i><font color="Green">Lemma39</font></i></a>, </i>;<br/><b>hence </b><a NAME="E14:28_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="rearran1.html#K3" target="_self">Rland</a> <font color="Maroon">F</font>,<font color="Maroon">B</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="rearran1.html#E2:28"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E23:28"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="rearran1.html#K3" target="_self">Rland</a> <font color="Maroon">F</font>,<font color="Maroon">B</font></span>)</span>
 ;<br/>


</div>
