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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="newton.html#NM1">Prime</a>;<br/>



<b>assume </b><a NAME="E1:54"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>,   <a href="hidden.html#M1">set</a> ] means ( ( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 implies <font color="Maroon">c<sub>2</sub></font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font> ) &amp; ( <font color="Maroon">c<sub>2</sub></font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 ) &amp; ( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 implies  not <font color="Maroon">c<sub>2</sub></font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font> ) &amp; ( not <font color="Maroon">c<sub>2</sub></font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 ) );<br/>
<a NAME="E2:54"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</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> &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>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>] holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font>
 
;<br/>
<a NAME="E3:54"/><i><font color="Green">E55</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="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 <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="54_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:54_1"/>
<font color="Maroon">c<sub>3</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/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font> or  not <font color="Maroon">c<sub>2</sub></font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1_1"><b>suppose </b></a><a NAME="E1:54_1_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>4</sub></font> = 1;<br/><b>take </b>
1
;<br/><b>thus </b><a NAME="E2:54_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>,1]
 <b>by </b><i><a class="txt" href="nat_4.html#E1:54_1_1_1"><i><font color="Green">E56</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1_2"><b>suppose </b></a><a NAME="E1:54_1_1_2"/><i><font color="Green">E56</font></i>: 
not <font color="Maroon">c<sub>2</sub></font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>4</sub></font> = 0;<br/><b>take </b>
0
;<br/><b>thus </b><a NAME="E2:54_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>,0]
 <b>by </b><i><a class="txt" href="nat_4.html#E1:54_1_1_2"><i><font color="Green">E56</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E5:54"/><i><font color="Green">E56</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</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>and </b><br/><a NAME="E6:54"/><i><font color="Green">E57</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="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>3</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="finseq_1.html#S1">FINSEQ_1:sch 1</a>(<a class="txt" href="nat_4.html#E2:54"><i><font color="Green">E54</font></i></a>, <a class="txt" href="nat_4.html#E3:54"><i><font color="Green">E55</font></i></a>);<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2"><b>now </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="E1:54_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:54_2"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</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> )
 <b>by </b><i><a class="txt" href="nat_4.html#E1:54_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="nat_4.html#E3:54_2"><i><font color="Green">E59</font></i></a></i>;<br/><a NAME="E5:54_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>]
 <b>by </b><i><a class="txt" href="nat_4.html#E3:54_2"><i><font color="Green">E59</font></i></a>, <a class="txt" href="nat_4.html#E5:54"><i><font color="Green">E56</font></i></a>, <a class="txt" href="nat_4.html#E6:54"><i><font color="Green">E57</font></i></a></i>;<br/><b>hence </b><a NAME="E6:54_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="txt" href="nat_4.html#E3:54_2"><i><font color="Green">E59</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:54"/><b>then </b><i><font color="Green">E58</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as    <a href="trees_4.html#M1">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="nat_4.html#E8:54"><i><font color="Green">E58</font></i></a>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<b>thus </b><a NAME="E10:54"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="nat_4.html#E5:54"><i><font color="Green">E56</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>

<b>thus </b><a NAME="E11:54"/>
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="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> holds <br/>( ( <font color="Maroon">c<sub>4</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 implies <font color="Maroon">c<sub>2</sub></font> <a href="wsierp_1.html#K2">|^</a> <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font> ) &amp; ( <font color="Maroon">c<sub>2</sub></font> <a href="wsierp_1.html#K2">|^</a> <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font> implies <font color="Maroon">c<sub>4</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 ) &amp; ( <font color="Maroon">c<sub>4</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 implies  not <font color="Maroon">c<sub>2</sub></font> <a href="wsierp_1.html#K2">|^</a> <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font> ) &amp; ( not <font color="Maroon">c<sub>2</sub></font> <a href="wsierp_1.html#K2">|^</a> <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font> implies <font color="Maroon">c<sub>4</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 ) )
 <b>by </b><i><a class="txt" href="nat_4.html#E5:54"><i><font color="Green">E56</font></i></a>, <a class="txt" href="nat_4.html#E6:54"><i><font color="Green">E57</font></i></a></i>;<br/>

<b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 1</span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<div><i><font color="Green">E59</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="54_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:54_3"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E2:54_3"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 <b>by </b><i><a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/><a NAME="E3:54_3"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="nat_4.html#E1:54"><i><font color="Green">E53</font></i></a>, <a class="ref" href="nat_3.html#D7">NAT_3:def 7</a></i>;<br/><a NAME="E4:54_3"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="nat_4.html#E1:54"><i><font color="Green">E53</font></i></a></i>;<br/><a NAME="E5:54_3"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="nat_4.html#E3:54_3"><i><font color="Green">E61</font></i></a>, <a class="ref" href="nat_1.html#T54">NAT_1:54</a></i>;<br/><a NAME="E6:54_3"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i><a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/><a NAME="E7:54_3"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="nat_4.html#E1:54_3"><i><font color="Green">E60</font></i></a>, <a class="ref" href="power.html#T44">POWER:44</a></i>;<br/><a NAME="E8:54_3"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T48">POWER:48</a></i>;<br/><a NAME="E9:54_3"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T48">POWER:48</a></i>;<br/><a NAME="E10:54_3"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>1</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="nat_4.html#E5:54_3"><i><font color="Green">E62</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>hence </b><a NAME="E11:54_3"/>
contradiction
 <b>by </b><i><a class="txt" href="nat_4.html#E6:54_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="nat_4.html#T3" target="_self">Th3</a></i>;<br/></div><b>end;</b></div>
<i><font color="Green">E60</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 1</span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_2.html#T69">FINSEQ_2:69</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_2.html#T69">FINSEQ_2:69</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="nat_4.html#E12:54"><i><font color="Green">E59</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
;<br/>
<a NAME="E14:54"/><b>then </b><i><font color="Green">E61</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 1</span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="nat_4.html#E5:54"><i><font color="Green">E56</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E15:54"/><i><font color="Green">E62</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="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K8">.</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"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 1</span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:54_5"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:54_5"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 1</span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="nat_4.html#E13:54"><i><font color="Green">E60</font></i></a>, <a class="txt" href="nat_4.html#E5:54"><i><font color="Green">E56</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 1;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 0;<br/>
<b>set </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>8</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="nat_4.html#E1:54_5"><i><font color="Green">E63</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_5_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 1</span>)</span> or 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#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Olive">b<sub>1</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="nat_4.html#E2:54_5"><i><font color="Green">E64</font></i></a>, <a class="ref" href="finseq_1.html#T38">FINSEQ_1:38</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_5_1_1"><b>suppose </b></a><a NAME="E1:54_5_1_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E2:54_5_1_1"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T68">FINSEQ_2:68</a></i>;<br/><a NAME="E3:54_5_1_1"/><b>then </b><i><font color="Green">E67</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 1</span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="finseq_2.html#T70">FINSEQ_2:70</a></i>;<br/><a NAME="E4:54_5_1_1"/><i><font color="Green">E68</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 1</span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="nat_4.html#E3:54_5_1_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="nat_4.html#E1:54_5_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E5:54_5_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_5_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_5_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="E1:54_5_1_1_1_1"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>12</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="54_5_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:54_5_1_1_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:54_5_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="newton.html#T9">NEWTON:9</a></i>;<br/><b>hence </b><a NAME="E3:54_5_1_1_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T53">NAT_1:53</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_5_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:54_5_1_1_1_1_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:54_5_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E3:54_5_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:54_5_1_1_1_1_2"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="binop_2.html#K10">-</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><b>set </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</a> 1;<br/><b>set </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E5:54_5_1_1_1_1_2"/><i><font color="Green">E70</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="nat_4.html#E4:54_5_1_1_1_1_2"><i><font color="Green">E69</font></i></a></i>;<br/><a NAME="E6:54_5_1_1_1_1_2"/><i><font color="Green">E71</font></i>: 
<span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="binop_2.html#K9">+</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 0 <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E7:54_5_1_1_1_1_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="nat_4.html#E2:54_5_1_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E8:54_5_1_1_1_1_2"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="nat_4.html#E4:54_5_1_1_1_1_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="nat_4.html#E6:54_5_1_1_1_1_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E9:54_5_1_1_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 <b>by </b><i><a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/><a NAME="E10:54_5_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="nat_4.html#E1:54"><i><font color="Green">E53</font></i></a>, <a class="ref" href="nat_3.html#D7">NAT_3:def 7</a></i>;<br/><b>hence </b><a NAME="E11:54_5_1_1_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="nat_4.html#E5:54_5_1_1_1_1_2"><i><font color="Green">E70</font></i></a>, <a class="txt" href="nat_4.html#E8:54_5_1_1_1_1_2"><i><font color="Green">E72</font></i></a>, <a class="ref" href="nat_3.html#T4">NAT_3:4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E6:54_5_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K8">.</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"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 1</span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="nat_4.html#E4:54_5_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="nat_4.html#E1:54_5"><i><font color="Green">E63</font></i></a>, <a class="txt" href="nat_4.html#E5:54"><i><font color="Green">E56</font></i></a>, <a class="txt" href="nat_4.html#E6:54"><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="54_5_1_2"><b>suppose </b></a><a NAME="E1:54_5_1_2"/><i><font color="Green">E65</font></i>: 
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#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/><div class="add"><b>consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:54_5_1_2"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="txt" href="nat_4.html#E1:54_5_1_2"><i><font color="Green">E65</font></i></a></i>;<br/><a NAME="E4:54_5_1_2"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="nat_4.html#E3:54_5_1_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="finseq_2.html#T68">FINSEQ_2:68</a></i>;<br/><a NAME="E5:54_5_1_2"/><b>then </b><i><font color="Green">E68</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="finseq_2.html#T70">FINSEQ_2:70</a></i>;<br/><a NAME="E6:54_5_1_2"/><i><font color="Green">E69</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 1</span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="nat_4.html#E5:54_5_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="nat_4.html#E3:54_5_1_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E7:54_5_1_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="nat_4.html#E4:54_5_1_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E8:54_5_1_2"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E9:54_5_1_2"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="nat_4.html#E3:54_5_1_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="finseq_2.html#T69">FINSEQ_2:69</a></i>;<br/><a NAME="E10:54_5_1_2"/>
not <font color="Maroon">c<sub>2</sub></font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_5_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:54_5_1_2_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:54_5_1_2_1"/><b>then </b><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="nat_4.html#E1:54"><i><font color="Green">E53</font></i></a>, <a class="ref" href="nat_3.html#T30">NAT_3:30</a></i>;<br/>
<a NAME="E3:54_5_1_2_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>
<b>hence </b><a NAME="E4:54_5_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="nat_4.html#E9:54_5_1_2"><i><font color="Green">E70</font></i></a>, <a class="txt" href="nat_4.html#E2:54_5_1_2_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="nat_3.html#T25">NAT_3:25</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E11:54_5_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K8">.</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"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 1</span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="nat_4.html#E6:54_5_1_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="nat_4.html#E1:54_5"><i><font color="Green">E63</font></i></a>, <a class="txt" href="nat_4.html#E5:54"><i><font color="Green">E56</font></i></a>, <a class="txt" href="nat_4.html#E6:54"><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>set </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 1;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 0;<br/>
<a NAME="E16:54"/><i><font color="Green">E63</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 1 is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="finseq_2.html#T77">FINSEQ_2:77</a></i>;<br/>
<a NAME="E17:54"/><i><font color="Green">E64</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 0 is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="finseq_2.html#T77">FINSEQ_2:77</a></i>;<br/>
 <a href="wsierp_1.html#K8">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 1</span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="nat_4.html#E16:54"><i><font color="Green">E63</font></i></a>, <a class="txt" href="nat_4.html#E17:54"><i><font color="Green">E64</font></i></a>, <a class="ref" href="rvsum_1.html#T105">RVSUM_1:105</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_2.html#K24">*</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T110">RVSUM_1:110</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T110">RVSUM_1:110</a></i>
;<br/>
<b>hence </b><a NAME="E19:54"/>
<font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K8">Sum</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="nat_4.html#E14:54"><i><font color="Green">E61</font></i></a>, <a class="txt" href="nat_4.html#E15:54"><i><font color="Green">E62</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


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