<?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="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><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="E1:35"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a> 
 ;<br/>

<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <font color="Maroon">c<sub>2</sub></font> <a href="finsop_1.html#K2">"**"</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p2">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<font color="Maroon">c<sub>2</sub></font> <a href="finsop_1.html#K2">"**"</a> <span class="p1">(<span class="default">0 <a href="finsop_1.html#K1">|-&gt;</a> <span class="p2">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> = 
<font color="Maroon">c<sub>2</sub></font> <a href="finsop_1.html#K2">"**"</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_2.html#T72">FINSEQ_2:72</a></i>
<br/>.= 
 <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="setwop_2.html#E1:35"><i><font color="Green">E17</font></i></a>, <a class="ref" href="finsop_1.html#T11">FINSOP_1:11</a></i>
;<br/>
<a NAME="E3:35"/><b>then </b><i><font color="Green">E18</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E4:35"/><i><font color="Green">E19</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:35_2"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="finsop_1.html#K2">"**"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p2">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>thus </b><font color="Maroon">c<sub>2</sub></font> <a href="finsop_1.html#K2">"**"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p2">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> = 
<font color="Maroon">c<sub>2</sub></font> <a href="finsop_1.html#K2">"**"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p3">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_2.html#T74">FINSEQ_2:74</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finsop_1.html#K2">"**"</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p3">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="setwop_2.html#E1:35"><i><font color="Green">E17</font></i></a>, <a class="ref" href="finsop_1.html#T5">FINSOP_1:5</a></i>
<br/>.= 
 <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="setwop_2.html#E1:35"><i><font color="Green">E17</font></i></a>, <a class="txt" href="setwop_2.html#E1:35_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="setwiseo.html#T23">SETWISEO:23</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E5:35"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="setwop_2.html#E3:35"><i><font color="Green">E18</font></i></a>, <a class="txt" href="setwop_2.html#E4:35"><i><font color="Green">E19</font></i></a>);<br/></i>
<b>hence </b><a NAME="E6:35"/>
<font color="Maroon">c<sub>2</sub></font> <a href="finsop_1.html#K2">"**"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p2">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>


</div>
