<?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>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#K4" target="_self">Rlor</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">b</font> =  <a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</font>;<br/>
<b>set </b><font color="Maroon">h</font> =  <a href="rfunct_3.html#K16">CHI</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</font></span>)</span>,<font color="Maroon">C</font>;<br/>
<b>assume </b><a NAME="E1:36"/><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/>

<a NAME="E2:36"/><b>then </b><i><font color="Green">E34</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">F</font>,<font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</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/>;<br/>
<a NAME="E3:36"/><i><font color="Green">E42</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> <span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</font></span>)</span>,<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> <span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</font></span>)</span>,<font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</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">B</font></span>)</span> &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="E4:36"/><i><font color="Green">E43</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">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">p</font> =  <a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">F</font>,<font color="Maroon">B</font> as  <a href="finset_1.html#V1">finite</a> <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">C</font>, <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="finset_1.html#T29">FINSET_1:29</a></i>;<br/>
<a NAME="E6:36"/><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> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</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">B</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1_1"/>
(  not <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> or <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> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_1"><b>case </b></a><a NAME="E1:36_1_1_1_1"/>
not <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 class="add"><a NAME="E2:36_1_1_1_1"/><b>then </b>
( <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="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">p</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="txt" href="rearran1.html#E1:36"><i><font color="Green">E33</font></i></a>, </i>;<br/><b>hence </b><a NAME="E3:36_1_1_1_1"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><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> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2"><b>case </b></a><a NAME="E1:36_1_1_1_2"/><i><font color="Green">E45</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/><div class="add"><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="E2:36_1_1_1_2"/><i><font color="Green">E49</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/><a NAME="E3:36_1_1_1_2"/><i><font color="Green">E51</font></i>: 
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="Olive">n</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="finseq_3.html#T27">FINSEQ_3:27</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="E5:36_1_1_1_2"/><i><font color="Green">E52</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="ref" href="rearran1.html#D2" target="_self">Def2</a>);<br/></i><b>consider </b><font color="Maroon">ma</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="E7:36_1_1_1_2"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">ma</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="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S6">NAT_1:sch 6</a>(<a class="txt" href="rearran1.html#E4"><i><font color="Green">Lemma39</font></i></a>, <a class="ref" href="rearran1.html#D2" target="_self">Def2</a>);<br/></i><a NAME="E8:36_1_1_1_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">mi</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font>
 <b>by </b><i>, <a class="txt" href="rearran1.html#E5"><i><font color="Green">Lemma41</font></i></a></i>;<br/><a NAME="E9:36_1_1_1_2"/><i><font color="Green">E56</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> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font> &amp; <font color="Maroon">ma</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#E5"><i><font color="Green">Lemma41</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E10:36_1_1_1_2"/><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/><b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i/>;<br/><a NAME="E13:36_1_1_1_2"/>
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="txt" href="rearran1.html#E4:36"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E14:36_1_1_1_2"/><b>then </b><i><font color="Green">E57</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>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E15:36_1_1_1_2"/>
<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#T45">XREAL_1:45</a></i>;<br/><a NAME="E16:36_1_1_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> <font color="Maroon">ma</font> &amp; <font color="Maroon">m1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">mi</font> )
 <b>by </b><i><a class="txt" href="rearran1.html#E3:36"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E17:36_1_1_1_2"/><b>then </b><i><font color="Green">E59</font></i>: 
(  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m1</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">ma</font> &amp;  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">ma</font> is <a href="finset_1.html#V1">finite</a> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/><a NAME="E18:36_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">ma</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m1</font></span>)</span> <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="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:36_1_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">ma</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m1</font></span>)</span> <a href="tarski.html#R1">c=</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="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></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="36_1_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:36_1_1_1_2_1_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">ma</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m1</font></span>)</span>
 ;<br/>

<a NAME="E2:36_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E62</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">ma</font> &amp; not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m1</font> &amp;  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">ma</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">l</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E4:36_1_1_1_2_1_1"/><i><font color="Green">E63</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> &amp; <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font> &amp; ( <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;</a> 1 or <font color="Maroon">m1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">l</font> ) )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E5:36_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon">mi</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E6:36_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">mi</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E7:36_1_1_1_2_1_1"/>
<font color="Maroon">l</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#E4:36"><i><font color="Green">E43</font></i></a>, <a class="txt" href="rearran1.html#E2:36_1_1_1_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:36_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">l</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="txt" href="rearran1.html#E2:36_1_1_1_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>reconsider </b><font color="Maroon">o</font> = <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">l</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1_1_2_1_1_1_1"/>
( <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ma</font> or <font color="Maroon">l</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">ma</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_1_1_1_1_1"><b>case </b></a><a NAME="E1:36_1_1_1_2_1_1_1_1_1"/>
<font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ma</font>
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="rearran1.html#E5"><i><font color="Green">Lemma41</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:36_1_1_1_2_1_1_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</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="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="rearran1.html#E3:36_1_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_1_1_1_1_2"><b>case </b></a><a NAME="E1:36_1_1_1_2_1_1_1_1_2"/>
<font color="Maroon">l</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">ma</font>
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">ma</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E2:36_1_1_1_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:36_1_1_1_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">o</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E5"><i><font color="Green">Lemma41</font></i></a>, <a class="txt" href="rearran1.html#E3:36_1_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="rfinseq.html#T32">RFINSEQ:32</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_1_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_1_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1_1_2_1_1_1_1_2_1_1"/>
( <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">mi</font> or <font color="Maroon">l</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">mi</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_1_1_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:36_1_1_1_2_1_1_1_1_2_1_1_1"/>
<font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">mi</font>
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_1_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</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>hence </b><a NAME="E3:36_1_1_1_2_1_1_1_1_2_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</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="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="rearran1.html#E3:36_1_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_1_1_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:36_1_1_1_2_1_1_1_1_2_1_1_2"/>
<font color="Maroon">l</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">mi</font>
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_1_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">mi</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">l</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:36_1_1_1_2_1_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">r</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">o</font>
 <b>by </b><i>, <a class="txt" href="rearran1.html#E3:36_1_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="rfinseq.html#T32">RFINSEQ:32</a></i>;<br/><a NAME="E4:36_1_1_1_2_1_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E5:36_1_1_1_2"><i><font color="Green">E52</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E5:36_1_1_1_2_1_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</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>hence </b><a NAME="E6:36_1_1_1_2_1_1_1_1_2_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</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="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="rearran1.html#E3:36_1_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:36_1_1_1_2_1_1_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</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="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E11:36_1_1_1_2_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</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="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>


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

<b>let </b><font color="Maroon">y</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="E2:36_1_1_1_2_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</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="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E3:36_1_1_1_2_1"/><b>then </b><i><font color="Green">E91</font></i>: 
( <font color="Maroon">y</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">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<b>reconsider </b><font color="Maroon">l</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E5:36_1_1_1_2_1"/><i><font color="Green">E92</font></i>: 
<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">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E6:36_1_1_1_2_1"/><b>then </b>
<font color="Maroon">mi</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:36_1_1_1_2_1"/><b>then </b>
<font color="Maroon">mi</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E8:36_1_1_1_2_1"/><b>then </b>
( <font color="Maroon">m1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">l</font> or <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;</a> 1 )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E9:36_1_1_1_2_1"/><b>then </b><i><font color="Green">E93</font></i>: 
not <font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m1</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E10:36_1_1_1_2_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> &amp; <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font> )
 
<b>by </b><i><a class="txt" href="rearran1.html#E5"><i><font color="Green">Lemma41</font></i></a>, , <a class="txt" href="rearran1.html#E2:36_1_1_1_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E11:36_1_1_1_2_1"/><b>then </b>
<font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">ma</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E12:36_1_1_1_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">ma</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m1</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div><b>then </b><i><font color="Green">E94</font></i>:  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><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> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">ma</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rearran1.html#E6"><i><font color="Green">Lemma46</font></i></a>, <a class="ref" href="card_2.html#T63">CARD_2:63</a></i>
<br/>.= 
<font color="Maroon">ma</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T78">FINSEQ_1:78</a></i>
<br/>.= 
<font color="Maroon">ma</font> <a href="real_1.html#K5">-</a> <font color="Maroon">m1</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T78">FINSEQ_1:78</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1_1_2_3_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="36_1_1_1_2_3_1_1"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_1"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">mi</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_3_1_1"/>
<span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">ma</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:36_1_1_1_2_3_1_1_1"/>
<span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">ma</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">p</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></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="36_1_1_1_2_3_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:36_1_1_1_2_3_1_1_1_1"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">y</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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">ma</font>
 ;<br/>

<a NAME="E2:36_1_1_1_2_3_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">ma</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="txt" href="rearran1.html#E2:36"><i><font color="Green">E34</font></i></a>, <a class="ref" href="rearran1.html#D1" target="_self">Def1</a>, <a class="txt" href="rearran1.html#E5"><i><font color="Green">Lemma41</font></i></a>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">cy</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="txt" href="rearran1.html#E2:36_1_1_1_2"><i><font color="Green">E49</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</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="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</font></span>)</span> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font> implies for <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font>  st <font color="Olive">c</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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> );<br/>
<a NAME="E4:36_1_1_1_2_3_1_1_1_1"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E5:36_1_1_1_2_3_1_1_1_1"/><i><font color="Green">E101</font></i>: 
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>2</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:36_1_1_1_2_3_1_1_1_1_1"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>]
 ;<br/>

<b>assume </b><a NAME="E2:36_1_1_1_2_3_1_1_1_1_1"/><i><font color="Green">E103</font></i>: 
( <font color="Maroon">n</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">B</font></span>)</span> &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font> )
 ;<br/>

<a NAME="E3:36_1_1_1_2_3_1_1_1_1_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">n</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">B</font></span>)</span> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</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="E4:36_1_1_1_2_3_1_1_1_1_1"/><b>then </b><i><font color="Green">E104</font></i>: 
( <font color="Maroon">n</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">B</font></span>)</span> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font> &amp; <font color="Maroon">n</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">B</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 &amp; <font color="Maroon">n</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">B</font></span>)</span> )
 
<b>by </b><i>, <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/>
<b>let </b><font color="Maroon">c</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font>;<br/>

<b>assume </b><a NAME="E5:36_1_1_1_2_3_1_1_1_1_1"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">c</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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1_1_2_3_1_1_1_1_1_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">n</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="36_1_1_1_2_3_1_1_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_1_1_1_1_1_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_3_1_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">p</font> <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="txt" href="rearran1.html#E8:36_1_1_1_2"><i><font color="Green">E55</font></i></a>, </i>;<br/><a NAME="E3:36_1_1_1_2_3_1_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</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>hence </b><a NAME="E4:36_1_1_1_2_3_1_1_1_1_1_1_1_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_1_1_1_1_1_1_2"/>
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_3_1_1_1_1_1_1_1_2"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E3:36_1_1_1_2_3_1_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E122</font></i>: 
( 0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; 0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E4:36_1_1_1_2_3_1_1_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E7:36_1_1_1_2"><i><font color="Green">E54</font></i></a>, </i>;<br/><b>then </b><i><font color="Green">E123</font></i>: <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xboole_0.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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K6">\</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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xboole_0.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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T39">XBOOLE_1:39</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_1_1_1_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1"/>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> or <font color="Maroon">c</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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 </span><b>by </b><i><a class="txt" href="rearran1.html#E8:36_1_1_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="rearran1.html#E14:36_1_1_1_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1"/><b>then </b><i><font color="Green">E124</font></i>: 
<font color="Maroon">p</font> <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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="txt" href="rearran1.html#E7:36_1_1_1_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="rearran1.html#E9:36_1_1_1_2"><i><font color="Green">E56</font></i></a>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1_1_1"/>
( <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">ma</font> or <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">ma</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1_1_1_1"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1_1_1_1"/>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">ma</font>
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="rearran1.html#E5"><i><font color="Green">Lemma41</font></i></a>, <a class="txt" href="rearran1.html#E16:36_1_1_1_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1_1_1_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1_1_1_2"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1_1_1_2"/>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">ma</font>
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">ma</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E125</font></i>: 
<font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E2:36"><i><font color="Green">E34</font></i></a>, <a class="ref" href="rearran1.html#D1" target="_self">Def1</a>, <a class="txt" href="rearran1.html#E5"><i><font color="Green">Lemma41</font></i></a>, , <a class="txt" href="rearran1.html#E16:36_1_1_1_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="rfinseq.html#T32">RFINSEQ:32</a></i>;<br/><a NAME="E4:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1_1_1_2"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E2:36"><i><font color="Green">E34</font></i></a>, <a class="ref" href="rearran1.html#D1" target="_self">Def1</a>, , , , <a class="txt" href="rearran1.html#E9:36_1_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="rearran1.html#E16:36_1_1_1_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="rfinseq.html#T32">RFINSEQ:32</a></i>;<br/><a NAME="E5:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E17:36_1_1_1_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E6:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</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>hence </b><a NAME="E7:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1_1_1_2"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_2"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_2"/>
<font color="Maroon">c</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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_2"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="rearran1.html#E5:36_1_1_1_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="rearran1.html#E7:36_1_1_1_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="rearran1.html#E9:36_1_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:36_1_1_1_2_3_1_1_1_1_1_1_1_2"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:36_1_1_1_2_3_1_1_1_1_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:36_1_1_1_2_3_1_1_1_1"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">ma</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">B</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rearran1.html#E2:36"><i><font color="Green">E34</font></i></a>, <a class="txt" href="rearran1.html#E5"><i><font color="Green">Lemma41</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E7:36_1_1_1_2_3_1_1_1_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, <a class="txt" href="rearran1.html#E3:36_1_1_1_2"><i><font color="Green">E51</font></i></a>);<br/></i>
<a NAME="E8:36_1_1_1_2_3_1_1_1_1"/><b>then </b>
<font color="Maroon">cy</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="rearran1.html#E2:36_1_1_1_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="rearran1.html#E5:36_1_1_1_2"><i><font color="Green">E52</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:36_1_1_1_2_3_1_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>


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

<b>let </b><font color="Maroon">y</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="E2:36_1_1_1_2_3_1_1_1"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">cy</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font> ;<br/>
<b>assume </b><a NAME="E4:36_1_1_1_2_3_1_1_1"/><i><font color="Green">E129</font></i>: 
not <font color="Maroon">y</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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">ma</font>
 ;<br/>

<a NAME="E5:36_1_1_1_2_3_1_1_1"/>
( <font color="Maroon">cy</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">p</font> &amp; <font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">cy</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="txt" href="rearran1.html#E2:36_1_1_1_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E6:36_1_1_1_2_3_1_1_1"/><b>then </b><i><font color="Green">E130</font></i>: 
<font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">cy</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</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="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</font></span>)</span> &amp; <font color="Maroon">ma</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">cy</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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E7:36_1_1_1_2_3_1_1_1"/><i><font color="Green">E131</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>2</sub></font>[<font color="Olive">n</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_2"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">n</font> =  <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">B</font></span>)</span>;<br/>

<a NAME="E1:36_1_1_1_2_3_1_1_1_2"/><i><font color="Green">E132</font></i>: 
<span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</font></span>)</span> <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/>
<a NAME="E2:36_1_1_1_2_3_1_1_1_2"/>
 <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">B</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E3:36_1_1_1_2_3_1_1_1_2"/><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="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</font></span>)</span>
 
;<br/>
<a NAME="E4:36_1_1_1_2_3_1_1_1_2"/><b>then </b>
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="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>hence </b><a NAME="E5:36_1_1_1_2_3_1_1_1_2"/>
<font color="Maroon">n</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">B</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_1_1_1_2_3_1_1_1_2_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font>
 ;<br/><a NAME="E2:36_1_1_1_2_3_1_1_1_2_1"/><b>then </b>
<font color="Maroon">ma</font> <a href="hidden.html#R1">=</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">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E2:36"><i><font color="Green">E34</font></i></a>, <a class="txt" href="rearran1.html#E4:36"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E3:36_1_1_1_2_3_1_1_1_2_1"/><b>then </b>
<font color="Maroon">cy</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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">ma</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:36_1_1_1_2_3_1_1_1_2_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:36_1_1_1_2_3_1_1_1_2"/>
( <font color="Maroon">ma</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> &amp; <font color="Maroon">cy</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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">mm</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="E9:36_1_1_1_2_3_1_1_1"/><i><font color="Green">E133</font></i>: 
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">mm</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>2</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">mm</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#E5:36_1_1_1_2"><i><font color="Green">E52</font></i></a>);<br/></i>
<a NAME="E10:36_1_1_1_2_3_1_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">mm</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E11:36_1_1_1_2_3_1_1_1"/><b>then </b>
 <a href="xxreal_0.html#K4">max</a> 0,<span class="p1">(<span class="default"><font color="Maroon">mm</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">mm</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">1m</font> = <font color="Maroon">mm</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/>
<a NAME="E13:36_1_1_1_2_3_1_1_1"/>
( <font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">mm</font> &amp; <font color="Maroon">mm</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">mm</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">mm</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">mm</font> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E14:36_1_1_1_2_3_1_1_1"/><b>then </b><i><font color="Green">E134</font></i>: 
( <font color="Maroon">ma</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">1m</font> &amp; <font color="Maroon">mm</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">B</font></span>)</span> &amp; <font color="Maroon">1m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">mm</font> &amp; <font color="Maroon">1m</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">mm</font> )
 
<b>by </b><i>, <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="E15:36_1_1_1_2_3_1_1_1"/><b>then </b><i><font color="Green">E135</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">1m</font> &amp; <font color="Maroon">1m</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">B</font></span>)</span> &amp; <font color="Maroon">1m</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">B</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="rearran1.html#E4:36"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E16:36_1_1_1_2_3_1_1_1"/><b>then </b><i><font color="Green">E136</font></i>: 
( <font color="Maroon">1m</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">B</font></span>)</span> &amp; <font color="Maroon">1m</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">B</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E17:36_1_1_1_2_3_1_1_1"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">mm</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="txt" href="rearran1.html#E2:36"><i><font color="Green">E34</font></i></a>, , <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1_1_2_3_1_1_1_3_1"/>
( <font color="Maroon">ma</font> <a href="hidden.html#R1">=</a> <font color="Maroon">1m</font> or <font color="Maroon">ma</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">1m</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_3_1_1"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_1_1_3_1_1"/>
<font color="Maroon">ma</font> <a href="hidden.html#R1">=</a> <font color="Maroon">1m</font>
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_3_1_1_1_3_1_1"/><b>then </b>
<font color="Maroon">cy</font> <a href="hidden.html#R2">in</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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">1m</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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">1m</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E3:36_1_1_1_2_3_1_1_1_3_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">cy</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> <font color="Maroon">mm</font>
 <b>by </b><i>, <a class="txt" href="rearran1.html#E8:36_1_1_1_2"><i><font color="Green">E55</font></i></a>, </i>;<br/><b>hence </b><a NAME="E4:36_1_1_1_2_3_1_1_1_3_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="rearran1.html#E5"><i><font color="Green">Lemma41</font></i></a>, <a class="txt" href="rearran1.html#E3:36_1_1_1_2"><i><font color="Green">E51</font></i></a>, , <a class="txt" href="rearran1.html#E14:36_1_1_1_2"><i><font color="Green">E57</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_3_1_2"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_1_1_3_1_2"/>
<font color="Maroon">ma</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">1m</font>
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_3_1_1_1_3_1_2"/><b>then </b><i><font color="Green">E153</font></i>: 
<font color="Maroon">ma</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">1m</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E7:36_1_1_1_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_3_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_1_1_1_2_3_1_1_1_3_1_2_1"/>
<font color="Maroon">cy</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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">1m</font>
 ;<br/><a NAME="E2:36_1_1_1_2_3_1_1_1_3_1_2_1"/><b>then </b>
<font color="Maroon">mm</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">1m</font>
 <b>by </b><i>, <a class="txt" href="rearran1.html#E9:36_1_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="rearran1.html#E16:36_1_1_1_2"><i><font color="Green">E58</font></i></a></i>;<br/><a NAME="E3:36_1_1_1_2_3_1_1_1_3_1_2_1"/><b>then </b>
<font color="Maroon">mm</font> <a href="real_1.html#K5">-</a> <font color="Maroon">1m</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i><a class="ref" href="real_2.html#T106">REAL_2:106</a></i>;<br/><b>hence </b><a NAME="E4:36_1_1_1_2_3_1_1_1_3_1_2_1"/>
contradiction
 ;<br/></div><b>end;</b></div><a NAME="E4:36_1_1_1_2_3_1_1_1_3_1_2"/><b>then </b>
<font color="Maroon">cy</font> <a href="hidden.html#R2">in</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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">1m</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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">1m</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E5:36_1_1_1_2_3_1_1_1_3_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">cy</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> <font color="Maroon">mm</font>
 <b>by </b><i>, <a class="txt" href="rearran1.html#E8:36_1_1_1_2"><i><font color="Green">E55</font></i></a>, </i>;<br/><b>hence </b><a NAME="E6:36_1_1_1_2_3_1_1_1_3_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="rearran1.html#E5"><i><font color="Green">Lemma41</font></i></a>, <a class="txt" href="rearran1.html#E3:36_1_1_1_2"><i><font color="Green">E51</font></i></a>, , <a class="txt" href="rearran1.html#E14:36_1_1_1_2"><i><font color="Green">E57</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E19:36_1_1_1_2_3_1_1_1"/>
contradiction
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:36_1_1_1_2_3_1_1"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><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> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E2:36"><i><font color="Green">E34</font></i></a>, <a class="txt" href="rearran1.html#E4:36"><i><font color="Green">E43</font></i></a>, , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_2"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_2"/><i><font color="Green">E154</font></i>: 
<font color="Maroon">mi</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_3_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="txt" href="rearran1.html#E4:36"><i><font color="Green">E43</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:36_1_1_1_2_3_1_2"/><b>then </b><i><font color="Green">E155</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m1</font> &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="txt" href="rearran1.html#E4:36"><i><font color="Green">E43</font></i></a>, <a class="txt" href="rearran1.html#E1:36_1_1_1_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:36_1_1_1_2_3_1_2"/><b>then </b><i><font color="Green">E156</font></i>: 
<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="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E2:36"><i><font color="Green">E34</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:36_1_1_1_2_3_1_2"/><b>then </b><i><font color="Green">E157</font></i>: 
<span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">ma</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E2:36"><i><font color="Green">E34</font></i></a>, <a class="ref" href="rearran1.html#D1" target="_self">Def1</a>, <a class="txt" href="rearran1.html#E5"><i><font color="Green">Lemma41</font></i></a>, <a class="txt" href="rearran1.html#E1:36_1_1_1_2"><i><font color="Green">E45</font></i></a>, </i>;<br/><a NAME="E6:36_1_1_1_2_3_1_2"/>
<span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">ma</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E2:36"><i><font color="Green">E34</font></i></a>, <a class="ref" href="rearran1.html#D1" target="_self">Def1</a>, <a class="txt" href="rearran1.html#E5"><i><font color="Green">Lemma41</font></i></a>, </i>;<br/><a NAME="E7:36_1_1_1_2_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">ma</font> 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/><b>then reconsider </b><font color="Maroon">bma</font> = <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">ma</font>, <font color="Maroon">bm1</font> = <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="rearran1.html#E3:36_1_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/><a NAME="E9:36_1_1_1_2_3_1_2"/>
<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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">ma</font></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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:36_1_1_1_2_3_1_2_1"/>
<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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">ma</font></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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">p</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></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="36_1_1_1_2_3_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:36_1_1_1_2_3_1_2_1_1"/><i><font color="Green">E160</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">ma</font></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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font></span>)</span>
 ;<br/>

<a NAME="E2:36_1_1_1_2_3_1_2_1_1"/><b>then </b><i><font color="Green">E161</font></i>: 
( <font color="Maroon">y</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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">ma</font> &amp; not <font color="Maroon">y</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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E3:36_1_1_1_2_3_1_2_1_1"/>
<span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">ma</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="txt" href="rearran1.html#E2:36"><i><font color="Green">E34</font></i></a>, <a class="ref" href="rearran1.html#D1" target="_self">Def1</a>, <a class="txt" href="rearran1.html#E5"><i><font color="Green">Lemma41</font></i></a>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">cy</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/>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</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="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</font></span>)</span> &amp; <font color="Maroon">mi</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font> implies for <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font>  st <font color="Olive">c</font> <a href="hidden.html#R2">in</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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font></span>)</span> holds <br/><font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> );<br/>
<a NAME="E5:36_1_1_1_2_3_1_2_1_1"/><i><font color="Green">E162</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E6:36_1_1_1_2_3_1_2_1_1"/><i><font color="Green">E163</font></i>: 
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>2</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:36_1_1_1_2_3_1_2_1_1_1"/><i><font color="Green">E164</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>]
 ;<br/>

<b>assume </b><a NAME="E2:36_1_1_1_2_3_1_2_1_1_1"/><i><font color="Green">E165</font></i>: 
( <font color="Maroon">n</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">B</font></span>)</span> &amp; <font color="Maroon">mi</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font> )
 ;<br/>

<a NAME="E3:36_1_1_1_2_3_1_2_1_1_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">n</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">B</font></span>)</span> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</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="E4:36_1_1_1_2_3_1_2_1_1_1"/><b>then </b><i><font color="Green">E166</font></i>: 
( <font color="Maroon">n</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">B</font></span>)</span> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font> &amp; <font color="Maroon">n</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">B</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 &amp; <font color="Maroon">n</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">B</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="rearran1.html#E14:36_1_1_1_2"><i><font color="Green">E57</font></i></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/>
<b>let </b><font color="Maroon">c</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font>;<br/>

<b>assume </b><a NAME="E5:36_1_1_1_2_3_1_2_1_1_1"/><i><font color="Green">E167</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1_1_2_3_1_2_1_1_1_1_1"/>
( <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">mi</font> or <font color="Maroon">mi</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_2_1_1_1_1_1_1"/><i><font color="Green">E168</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">mi</font>
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_3_1_2_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">p</font> <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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="txt" href="rearran1.html#E2:36_1_1_1_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="rearran1.html#E16:36_1_1_1_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="rearran1.html#E17:36_1_1_1_2"><i><font color="Green">E59</font></i></a>, </i>;<br/><a NAME="E3:36_1_1_1_2_3_1_2_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</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>hence </b><a NAME="E4:36_1_1_1_2_3_1_2_1_1_1_1_1_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_2_1_1_1_1_1_2"/>
<font color="Maroon">mi</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_3_1_2_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E169</font></i>: 
( <font color="Maroon">mi</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="rearran1.html#E14:36_1_1_1_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:36_1_1_1_2_3_1_2_1_1_1_1_1_2"/><b>then </b>
( <font color="Maroon">mi</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font> )
 <b>by </b><i><a class="txt" href="rearran1.html#E14:36_1_1_1_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:36_1_1_1_2_3_1_2_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E170</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</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">B</font></span>)</span> )
 <b>by </b><i><a class="txt" href="rearran1.html#E2:36"><i><font color="Green">E34</font></i></a>, <a class="txt" href="rearran1.html#E4:36"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:36_1_1_1_2_3_1_2_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">n</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">B</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:36_1_1_1_2_3_1_2_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E14:36_1_1_1_2"><i><font color="Green">E57</font></i></a>, , </i>;<br/><b>then </b><i><font color="Green">E171</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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font></span>)</span> = 
<span class="p1">(<span class="default"><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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xboole_0.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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xboole_0.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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T39">XBOOLE_1:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K6">\</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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T42">XBOOLE_1:42</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_1_1_1_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1"/>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K6">\</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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font></span>)</span> or <font color="Maroon">c</font> <a href="hidden.html#R2">in</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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font></span>)</span> )
 </span><b>by </b><i><a class="txt" href="rearran1.html#E17:36_1_1_1_2"><i><font color="Green">E59</font></i></a>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K6">\</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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E3:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1"/><b>then </b><i><font color="Green">E172</font></i>: 
<font color="Maroon">p</font> <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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="txt" href="rearran1.html#E16:36_1_1_1_2"><i><font color="Green">E58</font></i></a>, , </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1_1_1"/>
( <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">ma</font> or <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">ma</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1_1_1_1"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1_1_1_1"/>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">ma</font>
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="rearran1.html#E5"><i><font color="Green">Lemma41</font></i></a>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1_1_1_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1_1_1_2"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1_1_1_2"/>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">ma</font>
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">ma</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E14:36_1_1_1_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E173</font></i>: 
<font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E2:36"><i><font color="Green">E34</font></i></a>, <a class="ref" href="rearran1.html#D1" target="_self">Def1</a>, <a class="txt" href="rearran1.html#E5"><i><font color="Green">Lemma41</font></i></a>, <a class="txt" href="rearran1.html#E14:36_1_1_1_2"><i><font color="Green">E57</font></i></a>, , <a class="ref" href="rfinseq.html#T32">RFINSEQ:32</a></i>;<br/><a NAME="E4:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1_1_1_2"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E2:36"><i><font color="Green">E34</font></i></a>, <a class="ref" href="rearran1.html#D1" target="_self">Def1</a>, , <a class="txt" href="rearran1.html#E14:36_1_1_1_2"><i><font color="Green">E57</font></i></a>, , , <a class="ref" href="rfinseq.html#T32">RFINSEQ:32</a></i>;<br/><a NAME="E5:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E6:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</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>hence </b><a NAME="E7:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1_1_1_2"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_2"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_2"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_2"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="rearran1.html#E9:36_1_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="rearran1.html#E14:36_1_1_1_2"><i><font color="Green">E57</font></i></a>, , , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E9:36_1_1_1_2_3_1_2_1_1_1_1_1_2"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:36_1_1_1_2_3_1_2_1_1_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E7:36_1_1_1_2_3_1_2_1_1"/><i><font color="Green">E174</font></i>: 
<font color="Maroon">ma</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">B</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rearran1.html#E2:36"><i><font color="Green">E34</font></i></a>, <a class="txt" href="rearran1.html#E5"><i><font color="Green">Lemma41</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E8:36_1_1_1_2_3_1_2_1_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="rearran1.html#E7:36_1_1_1_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="rearran1.html#E8:36_1_1_1_2"><i><font color="Green">E55</font></i></a>);<br/></i>
<a NAME="E9:36_1_1_1_2_3_1_2_1_1"/><b>then </b>
<font color="Maroon">cy</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="rearran1.html#E3:36"><i><font color="Green">E42</font></i></a>, <a class="txt" href="rearran1.html#E5:36_1_1_1_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="rearran1.html#E9:36_1_1_1_2"><i><font color="Green">E56</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:36_1_1_1_2_3_1_2_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>


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

<b>let </b><font color="Maroon">y</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="E2:36_1_1_1_2_3_1_2_1"/><i><font color="Green">E175</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">cy</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font> ;<br/>
<a NAME="E4:36_1_1_1_2_3_1_2_1"/>
( <font color="Maroon">cy</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">p</font> &amp; <font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">cy</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="txt" href="rearran1.html#E5:36_1_1_1_2"><i><font color="Green">E52</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E5:36_1_1_1_2_3_1_2_1"/><b>then </b><i><font color="Green">E176</font></i>: 
<font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">cy</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>assume </b><a NAME="E6:36_1_1_1_2_3_1_2_1"/><i><font color="Green">E177</font></i>: 
not <font color="Maroon">y</font> <a href="hidden.html#R2">in</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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">ma</font></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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1_1_2_3_1_2_1_2_1"/>
(  not <font color="Maroon">y</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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">ma</font> or <font color="Maroon">y</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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font> )
 </span><b>by </b><i><a class="txt" href="rearran1.html#E7:36_1_1_1_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_2_1_1"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_2_1_2_1_1"/><i><font color="Green">E178</font></i>: 
not <font color="Maroon">y</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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">ma</font>
 ;<br/><div class="add"><b>defpred </b><font color="Maroon">S<sub>2</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="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</font></span>)</span> &amp; <font color="Maroon">ma</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">cy</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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/><a NAME="E2:36_1_1_1_2_3_1_2_1_2_1_1"/><i><font color="Green">E179</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>2</sub></font>[<font color="Olive">n</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_2_1_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> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</font></span>)</span>;<br/>

<a NAME="E1:36_1_1_1_2_3_1_2_1_2_1_1_1"/><i><font color="Green">E180</font></i>: 
<span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</font></span>)</span> <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/>
<a NAME="E2:36_1_1_1_2_3_1_2_1_2_1_1_1"/>
 <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">B</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E3:36_1_1_1_2_3_1_2_1_2_1_1_1"/><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="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</font></span>)</span>
 
;<br/>
<a NAME="E4:36_1_1_1_2_3_1_2_1_2_1_1_1"/><b>then </b>
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="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>hence </b><a NAME="E5:36_1_1_1_2_3_1_2_1_2_1_1_1"/>
<font color="Maroon">n</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">B</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_1_1_1_2_3_1_2_1_2_1_1_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font>
 ;<br/><a NAME="E2:36_1_1_1_2_3_1_2_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">ma</font> <a href="hidden.html#R1">=</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">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E2:36"><i><font color="Green">E34</font></i></a>, <a class="txt" href="rearran1.html#E4:36"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E3:36_1_1_1_2_3_1_2_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">cy</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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">ma</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E14:36_1_1_1_2"><i><font color="Green">E57</font></i></a></i>;<br/><b>hence </b><a NAME="E4:36_1_1_1_2_3_1_2_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="rearran1.html#E8:36_1_1_1_2"><i><font color="Green">E55</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:36_1_1_1_2_3_1_2_1_2_1_1_1"/>
( <font color="Maroon">ma</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> &amp; <font color="Maroon">cy</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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="txt" href="rearran1.html#E14:36_1_1_1_2"><i><font color="Green">E57</font></i></a></i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">mm</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="E4:36_1_1_1_2_3_1_2_1_2_1_1"/><i><font color="Green">E181</font></i>: 
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">mm</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>2</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">mm</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#E9:36_1_1_1_2"><i><font color="Green">E56</font></i></a>);<br/></i><a NAME="E5:36_1_1_1_2_3_1_2_1_2_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">mm</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E14:36_1_1_1_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:36_1_1_1_2_3_1_2_1_2_1_1"/><b>then </b>
 <a href="xxreal_0.html#K4">max</a> 0,<span class="p1">(<span class="default"><font color="Maroon">mm</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">mm</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">1m</font> = <font color="Maroon">mm</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/><a NAME="E8:36_1_1_1_2_3_1_2_1_2_1_1"/>
( <font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">mm</font> &amp; <font color="Maroon">mm</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">mm</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">mm</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">mm</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="rearran1.html#E14:36_1_1_1_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E9:36_1_1_1_2_3_1_2_1_2_1_1"/><b>then </b><i><font color="Green">E182</font></i>: 
( <font color="Maroon">ma</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">1m</font> &amp; <font color="Maroon">mm</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">B</font></span>)</span> &amp; <font color="Maroon">1m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">mm</font> &amp; <font color="Maroon">1m</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">mm</font> )
 <b>by </b><i><a class="txt" href="rearran1.html#E14:36_1_1_1_2"><i><font color="Green">E57</font></i></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="E10:36_1_1_1_2_3_1_2_1_2_1_1"/><b>then </b><i><font color="Green">E183</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">1m</font> &amp; <font color="Maroon">1m</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">B</font></span>)</span> &amp; <font color="Maroon">1m</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">B</font></span>)</span> )
 <b>by </b><i><a class="txt" href="rearran1.html#E4:36"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E11:36_1_1_1_2_3_1_2_1_2_1_1"/><b>then </b><i><font color="Green">E184</font></i>: 
( <font color="Maroon">1m</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">B</font></span>)</span> &amp; <font color="Maroon">1m</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">B</font></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E12:36_1_1_1_2_3_1_2_1_2_1_1"/><i><font color="Green">E185</font></i>: 
<font color="Maroon">mm</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="txt" href="rearran1.html#E2:36"><i><font color="Green">E34</font></i></a>, <a class="txt" href="rearran1.html#E14:36_1_1_1_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_2_1_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_2_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1_1_2_3_1_2_1_2_1_1_2_1"/>
( <font color="Maroon">ma</font> <a href="hidden.html#R1">=</a> <font color="Maroon">1m</font> or <font color="Maroon">ma</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">1m</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_2_1_1_2_1_1"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_2_1_2_1_1_2_1_1"/>
<font color="Maroon">ma</font> <a href="hidden.html#R1">=</a> <font color="Maroon">1m</font>
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_3_1_2_1_2_1_1_2_1_1"/><b>then </b>
<font color="Maroon">cy</font> <a href="hidden.html#R2">in</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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">1m</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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">1m</font></span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E8:36_1_1_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="rearran1.html#E14:36_1_1_1_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E3:36_1_1_1_2_3_1_2_1_2_1_1_2_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">cy</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> <font color="Maroon">mm</font>
 <b>by </b><i>, <a class="txt" href="rearran1.html#E17:36_1_1_1_2"><i><font color="Green">E59</font></i></a>, </i>;<br/><b>hence </b><a NAME="E4:36_1_1_1_2_3_1_2_1_2_1_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="rearran1.html#E5"><i><font color="Green">Lemma41</font></i></a>, , <a class="txt" href="rearran1.html#E14:36_1_1_1_2"><i><font color="Green">E57</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_2_1_1_2_1_2"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_2_1_2_1_1_2_1_2"/>
<font color="Maroon">ma</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">1m</font>
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_3_1_2_1_2_1_1_2_1_2"/><b>then </b><i><font color="Green">E186</font></i>: 
<font color="Maroon">ma</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">1m</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E16:36_1_1_1_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_2_1_1_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_1_1_1_2_3_1_2_1_2_1_1_2_1_2_1"/>
<font color="Maroon">cy</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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">1m</font>
 ;<br/><a NAME="E2:36_1_1_1_2_3_1_2_1_2_1_1_2_1_2_1"/><b>then </b>
<font color="Maroon">mm</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">1m</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E14:36_1_1_1_2"><i><font color="Green">E57</font></i></a>, , </i>;<br/><a NAME="E3:36_1_1_1_2_3_1_2_1_2_1_1_2_1_2_1"/><b>then </b>
<font color="Maroon">mm</font> <a href="real_1.html#K5">-</a> <font color="Maroon">1m</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i><a class="ref" href="real_2.html#T106">REAL_2:106</a></i>;<br/><b>hence </b><a NAME="E4:36_1_1_1_2_3_1_2_1_2_1_1_2_1_2_1"/>
contradiction
 ;<br/></div><b>end;</b></div><a NAME="E4:36_1_1_1_2_3_1_2_1_2_1_1_2_1_2"/><b>then </b>
<font color="Maroon">cy</font> <a href="hidden.html#R2">in</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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">1m</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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">1m</font></span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E14:36_1_1_1_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E5:36_1_1_1_2_3_1_2_1_2_1_1_2_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">cy</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> <font color="Maroon">mm</font>
 <b>by </b><i>, <a class="txt" href="rearran1.html#E17:36_1_1_1_2"><i><font color="Green">E59</font></i></a>, </i>;<br/><b>hence </b><a NAME="E6:36_1_1_1_2_3_1_2_1_2_1_1_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="rearran1.html#E5"><i><font color="Green">Lemma41</font></i></a>, , <a class="txt" href="rearran1.html#E14:36_1_1_1_2"><i><font color="Green">E57</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E14:36_1_1_1_2_3_1_2_1_2_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_2_1_2"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_2_1_2_1_2"/><i><font color="Green">E187</font></i>: 
<font color="Maroon">y</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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font>
 ;<br/><div class="add"><b>defpred </b><font color="Maroon">S<sub>2</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="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</font></span>)</span> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m1</font> &amp; <font color="Maroon">cy</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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/><a NAME="E2:36_1_1_1_2_3_1_2_1_2_1_2"/><i><font color="Green">E188</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>2</sub></font>[<font color="Olive">n</font>]
 <b>by </b><i>, <a class="txt" href="rearran1.html#E8:36_1_1_1_2"><i><font color="Green">E55</font></i></a></i>;<br/><b>consider </b><font color="Maroon">mm</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="E4:36_1_1_1_2_3_1_2_1_2_1_2"/><i><font color="Green">E189</font></i>: 
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">mm</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>2</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">mm</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#E9:36_1_1_1_2"><i><font color="Green">E56</font></i></a>);<br/></i><a NAME="E5:36_1_1_1_2_3_1_2_1_2_1_2"/><i><font color="Green">E190</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">mm</font> &amp; <font color="Maroon">mm</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">B</font></span>)</span> )
 <b>by </b><i><a class="txt" href="rearran1.html#E14:36_1_1_1_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:36_1_1_1_2_3_1_2_1_2_1_2"/><b>then </b>
 <a href="xxreal_0.html#K4">max</a> 0,<span class="p1">(<span class="default"><font color="Maroon">mm</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">mm</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">1m</font> = <font color="Maroon">mm</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/><a NAME="E8:36_1_1_1_2_3_1_2_1_2_1_2"/><i><font color="Green">E191</font></i>: 
<font color="Maroon">mm</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="txt" href="rearran1.html#E2:36"><i><font color="Green">E34</font></i></a>, <a class="txt" href="rearran1.html#E14:36_1_1_1_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_2_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_2_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1_1_2_3_1_2_1_2_1_2_1_1"/>
( <font color="Maroon">mm</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">mm</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="36_1_1_1_2_3_1_2_1_2_1_2_1_1_1"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_2_1_2_1_2_1_1_1"/>
<font color="Maroon">mm</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_3_1_2_1_2_1_2_1_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">cy</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#E14:36_1_1_1_2"><i><font color="Green">E57</font></i></a>, </i>;<br/><a NAME="E3:36_1_1_1_2_3_1_2_1_2_1_2_1_1_1"/><b>then </b>
<font color="Maroon">mi</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i>, <a class="txt" href="rearran1.html#E6:36"><i><font color="Green">E44</font></i></a>, </i>;<br/><b>hence </b><a NAME="E4:36_1_1_1_2_3_1_2_1_2_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="rearran1.html#E4:36"><i><font color="Green">E43</font></i></a>, , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_2_1_2_1_1_2"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_2_1_2_1_2_1_1_2"/>
<font color="Maroon">mm</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_3_1_2_1_2_1_2_1_1_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">mm</font> &amp; <font color="Maroon">mm</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">mm</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">mm</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">mm</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="rearran1.html#E16:36_1_1_1_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:36_1_1_1_2_3_1_2_1_2_1_2_1_1_2"/><b>then </b><i><font color="Green">E192</font></i>: 
( 1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">mm</font> &amp; <font color="Maroon">1m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">mm</font> &amp; <font color="Maroon">1m</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">mm</font> )
 <b>by </b><i><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="E4:36_1_1_1_2_3_1_2_1_2_1_2_1_1_2"/><b>then </b><i><font color="Green">E193</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">1m</font> &amp; <font color="Maroon">1m</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">B</font></span>)</span> &amp; <font color="Maroon">1m</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">B</font></span>)</span> &amp; <font color="Maroon">1m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m1</font> )
 <b>by </b><i><a class="txt" href="rearran1.html#E14:36_1_1_1_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="rearran1.html#E16:36_1_1_1_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E5:36_1_1_1_2_3_1_2_1_2_1_2_1_1_2"/><b>then </b>
<font color="Maroon">1m</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">B</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:36_1_1_1_2_3_1_2_1_2_1_2_1_1_2"/><b>then </b>
not <font color="Maroon">cy</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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">1m</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E14:36_1_1_1_2"><i><font color="Green">E57</font></i></a>, , </i>;<br/><a NAME="E7:36_1_1_1_2_3_1_2_1_2_1_2_1_1_2"/><b>then </b>
<font color="Maroon">cy</font> <a href="hidden.html#R2">in</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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">1m</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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">1m</font></span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E14:36_1_1_1_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E8:36_1_1_1_2_3_1_2_1_2_1_2_1_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">cy</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">1m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , </i>;<br/><a NAME="E9:36_1_1_1_2_3_1_2_1_2_1_2_1_1_2"/><b>then </b>
<font color="Maroon">mi</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">mm</font>
 <b>by </b><i>, , <a class="txt" href="rearran1.html#E17:36_1_1_1_2"><i><font color="Green">E59</font></i></a></i>;<br/><a NAME="E10:36_1_1_1_2_3_1_2_1_2_1_2_1_1_2"/><b>then </b>
<font color="Maroon">m1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m1</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E14:36_1_1_1_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E11:36_1_1_1_2_3_1_2_1_2_1_2_1_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m1</font> <a href="real_1.html#K5">-</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="E12:36_1_1_1_2_3_1_2_1_2_1_2_1_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><b>hence </b><a NAME="E13:36_1_1_1_2_3_1_2_1_2_1_2_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E10:36_1_1_1_2_3_1_2_1_2_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:36_1_1_1_2_3_1_2_1"/>
contradiction
 ;<br/>


</div><b>end;</b></div><b>hence </b> <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">bma</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">bm1</font></span>)</span>
<b>by </b><i><a class="txt" href="rearran1.html#E3:36_1_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="card_2.html#T63">CARD_2:63</a></i>
<br/>.= 
<font color="Maroon">ma</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">bm1</font></span>)</span>
<b>by </b><i><a class="txt" href="rearran1.html#E2:36"><i><font color="Green">E34</font></i></a>, <a class="txt" href="rearran1.html#E4:36"><i><font color="Green">E43</font></i></a>, </i>
<br/>.= 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><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> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i><a class="txt" href="rearran1.html#E2:36"><i><font color="Green">E34</font></i></a>, , <a class="txt" href="rearran1.html#E2:36_1_1_1_2"><i><font color="Green">E49</font></i></a>, </i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E21:36_1_1_1_2"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><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> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E2:36_1"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><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> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:36"/>
 <a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">F</font>,<font color="Maroon">B</font>, <a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">D</font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 <b>by </b><i><a class="ref" href="rfinseq.html#D1">RFINSEQ:def 1</a></i>;<br/>


</div>
