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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="trees_4.html#M1">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>





<i><font color="Green">E19</font></i>:  <a href="finseq_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="euler_2.html#K1" target="_self">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="euler_2.html#K1" target="_self">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="euler_2.html#K1" target="_self">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euler_2.html#D1" target="_self">Def1</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T6">NEWTON:6</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="euler_2.html#D1" target="_self">Def1</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<i><font color="Green">E20</font></i>:  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="euler_2.html#K1" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="euler_2.html#K1" target="_self">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T6">NEWTON:6</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="euler_2.html#D1" target="_self">Def1</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<i><font color="Green">E21</font></i>:  <a href="finseq_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="euler_2.html#K1" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="euler_2.html#K1" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="euler_2.html#K1" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euler_2.html#D1" target="_self">Def1</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T6">NEWTON:6</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<i><font color="Green">E22</font></i>:  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="euler_2.html#K1" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="euler_2.html#K1" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T6">NEWTON:6</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E5:20"/>
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#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="euler_2.html#K1" target="_self">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</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"><font color="Maroon">c<sub>1</sub></font> <a href="euler_2.html#K1" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_5"><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="E1:20_5"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>4</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>
 ;<br/>

<a NAME="E2:20_5"/><b>then </b><i><font color="Green">E24</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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<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="euler_2.html#E2:20_5"><i><font color="Green">E24</font></i></a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="euler_2.html#K1" target="_self">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="wsierp_1.html#K3">.</a> <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>1</sub></font> <a href="euler_2.html#K1" target="_self">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="euler_2.html#E2:20"><i><font color="Green">E20</font></i></a>, <a class="txt" href="euler_2.html#E1:20_5"><i><font color="Green">E23</font></i></a>, <a class="ref" href="euler_2.html#D1" target="_self">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="rvsum_1.html#T66">RVSUM_1:66</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="euler_2.html#E1:20_5"><i><font color="Green">E23</font></i></a>, <a class="ref" href="euler_2.html#D1" target="_self">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="euler_2.html#T9" target="_self">Th9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="euler_2.html#K1" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="rvsum_1.html#T66">RVSUM_1:66</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="euler_2.html#K1" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="euler_2.html#E4:20"><i><font color="Green">E22</font></i></a>, <a class="txt" href="euler_2.html#E1:20_5"><i><font color="Green">E23</font></i></a>, <a class="ref" href="euler_2.html#D1" target="_self">Def1</a></i>
;<br/>
<b>hence </b><a NAME="E5:20_5"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="euler_2.html#K1" target="_self">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="euler_2.html#K1" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:20"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="euler_2.html#K1" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="euler_2.html#K1" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="euler_2.html#E1:20"><i><font color="Green">E19</font></i></a>, <a class="txt" href="euler_2.html#E3:20"><i><font color="Green">E21</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div>
