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

<b>set </b><font color="Maroon" title="c1">A</font> = <font color="Maroon">F<sub>1</sub></font>();<br/>
<b>consider </b><font color="Maroon" title="c2">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E2:129_1"/><i><font color="Green" title="E86">A6</font></i>: 
<font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="ref" href="aofa_000.html#T79" target="_self" title="AOFA_000:th.79">Th79</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>for <font color="Olive" title="b1">I</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive" title="b1">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> $1 holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">I</font>];<br/>
<a NAME="E3:129_1"/>
<span class="p1">(<span class="default"><a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="ref" href="aofa_000.html#T18" target="_self" title="AOFA_000:th.18">Th18</a></i>;<br/>
<a NAME="E4:129_1"/><b>then </b><i><font color="Green" title="E87">A7</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<b>by </b><i><a class="txt" href="aofa_000.html#E1:129"><i><font color="Green" title="E81">A1</font></i></a></i>;<br/>
<div><i><font color="Green" title="E88">A8</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="129_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:129_1_1"/><i><font color="Green" title="E89">A9</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font>]
 ;<br/><b>thus </b><a NAME="E2:129_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="129_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">I</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<b>assume </b><a NAME="E1:129_1_1_1"/><i><font color="Green" title="E90">A10</font></i>: 
<font color="Maroon" title="c4">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="129_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="E1:129_1_1_1_1"/>
( <font color="Maroon" title="c4">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <font color="Maroon" title="c3">n</font> or <font color="Maroon" title="c4">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aofa_000.html#K12" title="AOFA_000:func.12">EmptyIns</a> <font color="Maroon">F<sub>1</sub></font>() or  ex <font color="Olive" title="b1">I1</font>, <font color="Olive" title="b2">I2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Maroon" title="c4">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">I1</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Olive" title="b2">I2</font> &amp; <font color="Olive" title="b1">I1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Olive" title="b2">I2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <font color="Maroon" title="c3">n</font> ) or  ex <font color="Olive" title="b1">C</font>, <font color="Olive" title="b2">I1</font>, <font color="Olive" title="b3">I2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Maroon" title="c4">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aofa_000.html#K14" title="AOFA_000:func.14">if-then-else</a> <font color="Olive" title="b1">C</font>,<font color="Olive" title="b2">I1</font>,<font color="Olive" title="b3">I2</font> &amp; <font color="Olive" title="b1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Olive" title="b2">I1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Olive" title="b3">I2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <font color="Maroon" title="c3">n</font> ) or  ex <font color="Olive" title="b1">C</font>, <font color="Olive" title="b2">J</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Maroon" title="c4">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Olive" title="b1">C</font>,<font color="Olive" title="b2">J</font> &amp; <font color="Olive" title="b1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Olive" title="b2">J</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <font color="Maroon" title="c3">n</font> ) )
 </span><b>by </b><i><a class="txt" href="aofa_000.html#E1:129_1_1_1"><i><font color="Green" title="E90">A10</font></i></a>, <a class="ref" href="aofa_000.html#T77" target="_self" title="AOFA_000:th.77">Th77</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="129_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:129_1_1_1_1_1"/>
<font color="Maroon" title="c4">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <font color="Maroon" title="c3">n</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:129_1_1_1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">I</font>]
 <b>by </b><i><a class="txt" href="aofa_000.html#E1:129_1_1"><i><font color="Green" title="E89">A9</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="129_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:129_1_1_1_1_2"/>
<font color="Maroon" title="c4">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aofa_000.html#K12" title="AOFA_000:func.12">EmptyIns</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/><div class="add"><b>hence </b><a NAME="E2:129_1_1_1_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">I</font>]
 <b>by </b><i><a class="txt" href="aofa_000.html#E2:129"><i><font color="Green" title="E82">A2</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="129_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:129_1_1_1_1_3"/>
 ex <font color="Olive" title="b1">I1</font>, <font color="Olive" title="b2">I2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Maroon" title="c4">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">I1</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Olive" title="b2">I2</font> &amp; <font color="Olive" title="b1">I1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Olive" title="b2">I2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <font color="Maroon" title="c3">n</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c5">I1</font>, <font color="Maroon" title="c6">I2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E3:129_1_1_1_1_3"/><i><font color="Green" title="E91">A11</font></i>: 
( <font color="Maroon" title="c4">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">I1</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Maroon" title="c6">I2</font> &amp; <font color="Maroon" title="c5">I1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Maroon" title="c6">I2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <font color="Maroon" title="c3">n</font> )
 ;<br/><a NAME="E4:129_1_1_1_1_3"/>
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">I1</font>] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c6">I2</font>] )
 <b>by </b><i><a class="txt" href="aofa_000.html#E1:129_1_1"><i><font color="Green" title="E89">A9</font></i></a>, <a class="txt" href="aofa_000.html#E3:129_1_1_1_1_3"><i><font color="Green" title="E91">A11</font></i></a></i>;<br/><b>hence </b><a NAME="E5:129_1_1_1_1_3"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">I</font>]
 <b>by </b><i><a class="txt" href="aofa_000.html#E3:129"><i><font color="Green" title="E83">A3</font></i></a>, <a class="txt" href="aofa_000.html#E3:129_1_1_1_1_3"><i><font color="Green" title="E91">A11</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="129_1_1_1_1_4"><b>suppose </b></a><a NAME="E1:129_1_1_1_1_4"/>
 ex <font color="Olive" title="b1">C</font>, <font color="Olive" title="b2">I1</font>, <font color="Olive" title="b3">I2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Maroon" title="c4">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aofa_000.html#K14" title="AOFA_000:func.14">if-then-else</a> <font color="Olive" title="b1">C</font>,<font color="Olive" title="b2">I1</font>,<font color="Olive" title="b3">I2</font> &amp; <font color="Olive" title="b1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Olive" title="b2">I1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Olive" title="b3">I2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <font color="Maroon" title="c3">n</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c5">C</font>, <font color="Maroon" title="c6">I1</font>, <font color="Maroon" title="c7">I2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E3:129_1_1_1_1_4"/><i><font color="Green" title="E91">A12</font></i>: 
( <font color="Maroon" title="c4">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aofa_000.html#K14" title="AOFA_000:func.14">if-then-else</a> <font color="Maroon" title="c5">C</font>,<font color="Maroon" title="c6">I1</font>,<font color="Maroon" title="c7">I2</font> &amp; <font color="Maroon" title="c5">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Maroon" title="c6">I1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Maroon" title="c7">I2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <font color="Maroon" title="c3">n</font> )
 ;<br/><a NAME="E4:129_1_1_1_1_4"/>
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">C</font>] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c6">I1</font>] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c7">I2</font>] )
 <b>by </b><i><a class="txt" href="aofa_000.html#E1:129_1_1"><i><font color="Green" title="E89">A9</font></i></a>, <a class="txt" href="aofa_000.html#E3:129_1_1_1_1_4"><i><font color="Green" title="E91">A12</font></i></a></i>;<br/><b>hence </b><a NAME="E5:129_1_1_1_1_4"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">I</font>]
 <b>by </b><i><a class="txt" href="aofa_000.html#E4:129"><i><font color="Green" title="E84">A4</font></i></a>, <a class="txt" href="aofa_000.html#E3:129_1_1_1_1_4"><i><font color="Green" title="E91">A12</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="129_1_1_1_1_5"><b>suppose </b></a><a NAME="E1:129_1_1_1_1_5"/>
 ex <font color="Olive" title="b1">C</font>, <font color="Olive" title="b2">J</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Maroon" title="c4">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Olive" title="b1">C</font>,<font color="Olive" title="b2">J</font> &amp; <font color="Olive" title="b1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Olive" title="b2">J</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <font color="Maroon" title="c3">n</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c5">C</font>, <font color="Maroon" title="c6">J</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E3:129_1_1_1_1_5"/><i><font color="Green" title="E91">A13</font></i>: 
( <font color="Maroon" title="c4">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Maroon" title="c5">C</font>,<font color="Maroon" title="c6">J</font> &amp; <font color="Maroon" title="c5">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Maroon" title="c6">J</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <font color="Maroon" title="c3">n</font> )
 ;<br/><a NAME="E4:129_1_1_1_1_5"/>
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">C</font>] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c6">J</font>] )
 <b>by </b><i><a class="txt" href="aofa_000.html#E1:129_1_1"><i><font color="Green" title="E89">A9</font></i></a>, <a class="txt" href="aofa_000.html#E3:129_1_1_1_1_5"><i><font color="Green" title="E91">A13</font></i></a></i>;<br/><b>hence </b><a NAME="E5:129_1_1_1_1_5"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">I</font>]
 <b>by </b><i><a class="txt" href="aofa_000.html#E5:129"><i><font color="Green" title="E85">A5</font></i></a>, <a class="txt" href="aofa_000.html#E3:129_1_1_1_1_5"><i><font color="Green" title="E91">A13</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E6:129_1"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="aofa_000.html#E4:129_1"><i><font color="Green" title="E87">A7</font></i></a>, <a class="txt" href="aofa_000.html#E5:129_1"><i><font color="Green" title="E88">A8</font></i></a>);<br/></i>
<b>hence </b><a NAME="E7:129_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>2</sub></font>()]
 <b>by </b><i><a class="txt" href="aofa_000.html#E2:129_1"><i><font color="Green" title="E86">A6</font></i></a></i>;<br/>


</div>
