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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:86"/><i><font color="Green">E69</font></i>: 
( not <font color="Maroon">c<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; not <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a> )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="funct_2.html#V2">onto</a> </span>}</span> ;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K4">INT</a> <b> such that </b><br/><a NAME="E3:86"/><i><font color="Green">E70</font></i>: 
 <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E4:86"/><i><font color="Green">E71</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="card_fin.html#K6" target="_self">Sum</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="funct_2.html#V2">onto</a> </span>}</span> 
 <b>and </b><br/><a NAME="E5:86"/><i><font color="Green">E72</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="card_fin.html#K7" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="newton.html#K3">|^</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="newton.html#K9">choose</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="card_fin.html#E1:86"><i><font color="Green">E69</font></i></a>, <a class="txt" href="card_fin.html#E80"><i><font color="Green">Lemma67</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="card_fin.html#K7" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E6:86"/><i><font color="Green">E73</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:86_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:86_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="card_fin.html#K7" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>] &amp; <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="card_fin.html#K7" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  )
 
<b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:86_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K4">INT</a> <b> such that </b><br/><a NAME="E8:86"/><i><font color="Green">E74</font></i>: 
 <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E9:86"/><i><font color="Green">E75</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="card_fin.html#K7" target="_self">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="stirl2_1.html#S6">STIRL2_1:sch 6</a>(<a class="txt" href="card_fin.html#E6:86"><i><font color="Green">E73</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="newton.html#K3">|^</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1"><a href="afinsq_1.html#K9">&lt;%</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="afinsq_1.html#K9">%&gt;</a></span> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">c<sub>5</sub></font> as   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K4">INT</a>  ;<br/>
<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

<a NAME="E12:86"/>
(  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="afinsq_1.html#K1">len</a> <span class="p1"><a href="afinsq_1.html#K9">&lt;%</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="afinsq_1.html#K9">%&gt;</a></span> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="txt" href="card_fin.html#E8:86"><i><font color="Green">E74</font></i></a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a>, <a class="ref" href="afinsq_1.html#T37">AFINSQ_1:37</a></i>;<br/>
<b>hence </b><a NAME="E13:86"/><i><font color="Green">E76</font></i>: 
 <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="ref" href="afinsq_1.html#D4">AFINSQ_1:def 4</a></i>;<br/>

<a NAME="E14:86"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="card_fin.html#K7" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="card_fin.html#K7" target="_self">.</a> <font color="Olive">b<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="card_fin.html#E3:86"><i><font color="Green">E70</font></i></a>, <a class="txt" href="card_fin.html#E9:86"><i><font color="Green">E75</font></i></a></i>;<br/>
<a NAME="E15:86"/><b>then </b>
( <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="card_fin.html#K6" target="_self">Sum</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_fin.html#K6" target="_self">Sum</a> <font color="Maroon">c<sub>5</sub></font> &amp; <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="card_fin.html#K6" target="_self">Sum</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><a href="card_fin.html#K6" target="_self">Sum</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="card_fin.html#E3:86"><i><font color="Green">E70</font></i></a>, <a class="txt" href="card_fin.html#E8:86"><i><font color="Green">E74</font></i></a>, <a class="ref" href="card_fin.html#T64" target="_self">Th64</a></i>;<br/>
<b>then </b><font color="Maroon">c<sub>6</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="card_fin.html#K6" target="_self">Sum</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
<font color="Maroon">c<sub>6</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="card_fin.html#K6" target="_self">Sum</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>

<br/>.= 
<a href="binop_2.html#K44">addint</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><a href="card_fin.html#K6" target="_self">Sum</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="gr_cy_1.html#T14">GR_CY_1:14</a></i>
<br/>.= 
<a href="binop_2.html#K44">addint</a>  <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="binop_2.html#K44">addint</a>  <a href="stirl2_1.html#K4">"**"</a> <span class="p2"><a href="afinsq_1.html#K9">&lt;%</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="afinsq_1.html#K9">%&gt;</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="card_fin.html#K6" target="_self">Sum</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="stirl2_1.html#T44">STIRL2_1:44</a></i>
<br/>.= 
<a href="binop_2.html#K44">addint</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="binop_2.html#K44">addint</a>  <a href="stirl2_1.html#K4">"**"</a> <span class="p2"><a href="afinsq_1.html#K9">&lt;%</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="afinsq_1.html#K9">%&gt;</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="binop_2.html#K44">addint</a>  <a href="stirl2_1.html#K4">"**"</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>

<br/>.= 
<a href="binop_2.html#K44">addint</a>  <a href="stirl2_1.html#K4">"**"</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="stirl2_1.html#T47">STIRL2_1:47</a></i>
<br/>.= 
 <a href="card_fin.html#K6" target="_self">Sum</a> <font color="Maroon">c<sub>7</sub></font>

;<br/>
<b>hence </b><a NAME="E17:86"/>
 <a href="card_fin.html#K6" target="_self">Sum</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="funct_2.html#V2">onto</a> </span>}</span> 
 <b>by </b><i><a class="txt" href="card_fin.html#E4:86"><i><font color="Green">E71</font></i></a></i>;<br/>

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

<b>assume </b><a NAME="E18:86"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:86_3_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_3_1_1"><b>suppose </b></a><a NAME="E1:86_3_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:86_3_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="newton.html#K3">|^</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="card_fin.html#K7" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="newton.html#T9">NEWTON:9</a>, <a class="ref" href="newton.html#T29">NEWTON:29</a>, <a class="ref" href="afinsq_1.html#T39">AFINSQ_1:39</a></i>;<br/><b>hence </b><a NAME="E3:86_3_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="card_fin.html#K7" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_3_1_2"><b>suppose </b></a><a NAME="E1:86_3_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K10">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="nat_1.html#T60">NAT_1:60</a></i>;<br/><a NAME="E3:86_3_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="txt" href="card_fin.html#E13:86"><i><font color="Green">E76</font></i></a>, <a class="txt" href="card_fin.html#E18:86"><i><font color="Green">E77</font></i></a>, <a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/><a NAME="E4:86_3_1_2"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:86_3_1_2"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E6:86_3_1_2"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="afinsq_1.html#K1">len</a> <span class="p1"><a href="afinsq_1.html#K9">&lt;%</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="afinsq_1.html#K9">%&gt;</a></span> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="txt" href="card_fin.html#E8:86"><i><font color="Green">E74</font></i></a>, <a class="ref" href="euler_1.html#T1">EULER_1:1</a>, <a class="ref" href="afinsq_1.html#T37">AFINSQ_1:37</a></i>;<br/><a NAME="E7:86_3_1_2"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font> <a href="card_fin.html#K7" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="card_fin.html#K7" target="_self">.</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="card_fin.html#E8:86"><i><font color="Green">E74</font></i></a>, <a class="ref" href="afinsq_1.html#D4">AFINSQ_1:def 4</a></i>;<br/><a NAME="E8:86_3_1_2"/><b>then </b><i><font color="Green">E78</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="card_fin.html#K7" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="card_fin.html#K7" target="_self">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="card_fin.html#K7" target="_self">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="newton.html#K9">choose</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="card_fin.html#E3:86"><i><font color="Green">E70</font></i></a>, <a class="txt" href="card_fin.html#E5:86"><i><font color="Green">E72</font></i></a>, <a class="txt" href="card_fin.html#E9:86"><i><font color="Green">E75</font></i></a></i>;<br/><a NAME="E9:86_3_1_2"/><b>then </b><i><font color="Green">E79</font></i>: 
<span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>;<br/><a NAME="E10:86_3_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="card_fin.html#K7" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="card_fin.html#E8:86_3_1_2"><i><font color="Green">E78</font></i></a></i>;<br/><a NAME="E11:86_3_1_2"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="card_fin.html#K7" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 ;<br/><b>hence </b><a NAME="E12:86_3_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="card_fin.html#K7" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="card_fin.html#E9:86_3_1_2"><i><font color="Green">E79</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="E20:86"/>
<font color="Maroon">c<sub>7</sub></font> <a href="card_fin.html#K7" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 ;<br/>


</div>
