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<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>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="stirl2_1.html#K3" target="_self">block</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K9">choose</a> 2;<br/>
<a NAME="E1:81"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1"><b>proof </b></a><div class="add">

<a NAME="E1:81_1"/>
( 1 <a href="stirl2_1.html#K3" target="_self">block</a> 0 <a href="hidden.html#R1">=</a> 0 &amp; 1 <a href="newton.html#K9">choose</a> 2 <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="stirl2_1.html#T31" target="_self">Th31</a>, <a class="ref" href="newton.html#D3">NEWTON:def 3</a></i>;<br/>
<b>hence </b><a NAME="E2:81_1"/>
<font color="Maroon">S<sub>1</sub></font>[0]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:81"/><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> 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="binop_2.html#K23">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:81_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>]
 ;<br/>

<b>set </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1;<br/>
<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="stirl2_1.html#K3" target="_self">block</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="stirl2_1.html#K3" target="_self">block</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></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>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="stirl2_1.html#K3" target="_self">block</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="stirl2_1.html#T56" target="_self">Th56</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="binop_2.html#K23">+</a> 1</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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K9">choose</a> 2</span>)</span>
<b>by </b><i><a class="txt" href="stirl2_1.html#E1:81_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="stirl2_1.html#T26" target="_self">Th26</a>, <a class="ref" href="card_2.html#T20">CARD_2:20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K9">+</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="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> 2</span>)</span>
<b>by </b><i><a class="ref" href="stirl2_1.html#T61" target="_self">Th61</a></i>
<br/>.= 
<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="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> 2

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K9">choose</a> 2
<b>by </b><i><a class="ref" href="stirl2_1.html#T61" target="_self">Th61</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E3:81"/>
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="stirl2_1.html#E1:81"><i><font color="Green">E53</font></i></a>, <a class="txt" href="stirl2_1.html#E2:81"><i><font color="Green">E54</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:81"/>
<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="stirl2_1.html#K3" target="_self">block</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</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="newton.html#K9">choose</a> 2
 ;<br/>


</div>
