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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> = <span class="p1">{<span class="default"> <span class="p2"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> ;<br/>
<a NAME="E1:34"/><i><font color="Green">E31</font></i>: 
<span class="p1">{<span class="default"> <span class="p2"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> 
 
<b>by </b><i><a class="ref" href="uniroots.html#T29" target="_self">Th29</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 ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p5">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p5">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span> );<br/>
<a NAME="E2:34"/>
<span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> 0</span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> 0</span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">{<span class="default"> <span class="p2"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E4:34"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <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="34_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:34_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="uniroots.html#E1:34_1"><i><font color="Green">E33</font></i></a></i>;<br/>
<a NAME="E3:34_1"/><i><font color="Green">E34</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="uniroots.html#E1:34_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:34_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E5:34_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> 1
 
<b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<a NAME="E6:34_1"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="uniroots.html#E3:34_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E8:34_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="binarith.html#K5">-'</a> 1
 ;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span>;<br/>
<a NAME="E9:34_1"/>
<span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="uniroots.html#E6:34_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="uniroots.html#E8:34_1"><i><font color="Green">E36</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:34_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font>] )
 <b>by </b><i><a class="txt" href="uniroots.html#E8:34_1"><i><font color="Green">E36</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E6:34"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</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>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>(<a class="txt" href="uniroots.html#E4:34"><i><font color="Green">E32</font></i></a>);<br/></i>
<a NAME="E7:34"/><i><font color="Green">E34</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E8:34"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:34_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:34_2"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="uniroots.html#E1:34_2"><i><font color="Green">E35</font></i></a></i>;<br/>
<a NAME="E4:34_2"/>
0 <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E5:34_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="uniroots.html#E3:34_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/>
<a NAME="E6:34_2"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E8:34_2"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span> )
 <b>by </b><i><a class="txt" href="uniroots.html#E6:34"><i><font color="Green">E33</font></i></a></i>;<br/>
<a NAME="E9:34_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>;<br/>
<b>hence </b><a NAME="E10:34_2"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="uniroots.html#E3:34_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="uniroots.html#E6:34_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="uniroots.html#E8:34_2"><i><font color="Green">E38</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:34"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="funct_2.html#T16">FUNCT_2:16</a></i>;<br/>
<a NAME="E10:34"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:34_3"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/>

<a NAME="E2:34_3"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="uniroots.html#E1:34_3"><i><font color="Green">E36</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:34_3"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span> )
 <b>by </b><i><a class="txt" href="uniroots.html#E6:34"><i><font color="Green">E33</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:34_3"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span> )
 <b>by </b><i><a class="txt" href="uniroots.html#E6:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="uniroots.html#E2:34_3"><i><font color="Green">E37</font></i></a></i>;<br/>
<a NAME="E7:34_3"/><i><font color="Green">E40</font></i>: 
(  <a href="sin_cos.html#K24">cos</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K24">cos</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="sin_cos.html#K21">sin</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K21">sin</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="uniroots.html#E1:34_3"><i><font color="Green">E36</font></i></a>, <a class="txt" href="uniroots.html#E4:34_3"><i><font color="Green">E38</font></i></a>, <a class="txt" href="uniroots.html#E6:34_3"><i><font color="Green">E39</font></i></a>, <a class="ref" href="comptrig.html#T8">COMPTRIG:8</a></i>;<br/>
<a NAME="E8:34_3"/><i><font color="Green">E41</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="uniroots.html#E2:34_3"><i><font color="Green">E37</font></i></a>, <a class="txt" href="uniroots.html#E4:34_3"><i><font color="Green">E38</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E9:34_3"/><i><font color="Green">E42</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="uniroots.html#E2:34_3"><i><font color="Green">E37</font></i></a>, <a class="txt" href="uniroots.html#E6:34_3"><i><font color="Green">E39</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E10:34_3"/>
<font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T148">XREAL_1:148</a></i>;<br/>
<a NAME="E11:34_3"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="uniroots.html#E8:34_3"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E12:34_3"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="uniroots.html#E8:34_3"><i><font color="Green">E41</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<a NAME="E13:34_3"/>
<font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T148">XREAL_1:148</a></i>;<br/>
<a NAME="E14:34_3"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="uniroots.html#E9:34_3"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E15:34_3"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="uniroots.html#E9:34_3"><i><font color="Green">E42</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E16:34_3"/><i><font color="Green">E45</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="comptrig.html#T21">COMPTRIG:21</a>, <a class="ref" href="real_2.html#T121">REAL_2:121</a></i>;<br/>
<a NAME="E17:34_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="uniroots.html#E12:34_3"><i><font color="Green">E43</font></i></a>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<a NAME="E18:34_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>;<br/>
<a NAME="E19:34_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="comptrig.html#T21">COMPTRIG:21</a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E20:34_3"/><b>then </b><i><font color="Green">E46</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a>  )
 
<b>by </b><i><a class="txt" href="uniroots.html#E16:34_3"><i><font color="Green">E45</font></i></a>, <a class="ref" href="real_2.html#T125">REAL_2:125</a>, <a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>;<br/>
<a NAME="E21:34_3"/><i><font color="Green">E47</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="comptrig.html#T21">COMPTRIG:21</a>, <a class="ref" href="real_2.html#T121">REAL_2:121</a></i>;<br/>
<a NAME="E22:34_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="uniroots.html#E15:34_3"><i><font color="Green">E44</font></i></a>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<a NAME="E23:34_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>;<br/>
<a NAME="E24:34_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="comptrig.html#T21">COMPTRIG:21</a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E25:34_3"/><b>then </b>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a>  )
 
<b>by </b><i><a class="txt" href="uniroots.html#E21:34_3"><i><font color="Green">E47</font></i></a>, <a class="ref" href="real_2.html#T125">REAL_2:125</a>, <a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>;<br/>
<a NAME="E26:34_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="uniroots.html#E7:34_3"><i><font color="Green">E40</font></i></a>, <a class="txt" href="uniroots.html#E20:34_3"><i><font color="Green">E46</font></i></a>, <a class="ref" href="complex2.html#T12">COMPLEX2:12</a></i>;<br/>
<a NAME="E27:34_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/>
<a NAME="E28:34_3"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/>
<a NAME="E29:34_3"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="binarith.html#K5">-'</a> 1
 
<b>by </b><i><a class="ref" href="comptrig.html#T21">COMPTRIG:21</a>, <a class="ref" href="xcmplx_1.html#T5">XCMPLX_1:5</a></i>;<br/>
<a NAME="E30:34_3"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="txt" href="uniroots.html#E8:34_3"><i><font color="Green">E41</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/>
<b>hence </b><a NAME="E31:34_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="uniroots.html#E4:34_3"><i><font color="Green">E38</font></i></a>, <a class="txt" href="uniroots.html#E6:34_3"><i><font color="Green">E39</font></i></a>, <a class="txt" href="uniroots.html#E9:34_3"><i><font color="Green">E42</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:34"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E12:34"/><b>then </b>
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="tarski.html#R2">are_equipotent</a> 
 
<b>by </b><i><a class="txt" href="uniroots.html#E7:34"><i><font color="Green">E34</font></i></a>, <a class="ref" href="card_1.html#T60">CARD_1:60</a></i>;<br/>
<a NAME="E13:34"/><b>then </b>
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>, <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R2">are_equipotent</a> 
 
<b>by </b><i><a class="txt" href="uniroots.html#E7:34"><i><font color="Green">E34</font></i></a>, <a class="ref" href="relat_1.html#T146">RELAT_1:146</a></i>;<br/>
<a NAME="E14:34"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">{<span class="default"> <span class="p2"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="uniroots.html#E9:34"><i><font color="Green">E35</font></i></a>, <a class="ref" href="card_1.html#T21">CARD_1:21</a></i>;<br/>
<b>hence </b><a NAME="E15:34"/>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="uniroots.html#E1:34"><i><font color="Green">E31</font></i></a>, <a class="ref" href="finseq_1.html#T78">FINSEQ_1:78</a></i>;<br/>


</div>
