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

<a NAME="E1:3_1"/><i><font color="Green">E8</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">F<sub>4</sub></font>() &amp; <font color="Maroon">F<sub>4</sub></font>() <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">F<sub>3</sub></font>() )
 
<b>by </b><i><a class="txt" href="ami_4.html#E1:3"><i><font color="Green">E3</font></i></a>, <a class="txt" href="ami_4.html#E2:3"><i><font color="Green">E4</font></i></a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1">Nat</a>,   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>  <a href="subset_1.html#M1">M2</a>( <a href="numbers.html#K5">NAT</a> ) =  <a href="cqc_lang.html#K2">IFEQ</a> <font color="Maroon">a<sub>2</sub></font>,0,0,<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">a<sub>1</sub></font>);<br/>
<b>consider </b><font color="Maroon">c<sub>1</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:3_1"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>()
 <b>and </b><br/><a NAME="E4:3_1"/><i><font color="Green">E10</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="recdef_1.html#S4">RECDEF_1:sch 4</a>();<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>  <a href="subset_1.html#M1">M2</a>( <a href="numbers.html#K5">NAT</a> ) = <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
 <a href="cqc_lang.html#K2">IFEQ</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> 0</span>)</span>,0,0,<font color="Maroon">F<sub>2</sub></font>(0)
<b>by </b><i><a class="txt" href="ami_4.html#E4:3_1"><i><font color="Green">E10</font></i></a></i>
<br/>.= 
<font color="Maroon">F<sub>2</sub></font>(0)
<b>by </b><i><a class="txt" href="ami_4.html#E2:3"><i><font color="Green">E4</font></i></a>, <a class="txt" href="ami_4.html#E3:3_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
;<br/>
<a NAME="E6:3_1"/><b>then </b><i><font color="Green">E11</font></i>: 
( <font color="Maroon">H<sub>2</sub></font>(0) <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>() &amp; <font color="Maroon">H<sub>2</sub></font>(1) <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>() )
 
<b>by </b><i><a class="txt" href="ami_4.html#E4:3"><i><font color="Green">E6</font></i></a>, <a class="txt" href="ami_4.html#E3:3_1"><i><font color="Green">E9</font></i></a></i>;<br/>
<a NAME="E7:3_1"/><i><font color="Green">E12</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/><font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_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="E1:3_1_2"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:3_1_2_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_2_1_1_1"><b>case </b></a><a NAME="E1:3_1_2_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:3_1_2_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>2</sub></font>)
 <b>by </b><i><a class="txt" href="ami_4.html#E3:3"><i><font color="Green">E5</font></i></a>, <a class="txt" href="ami_4.html#E3:3_1"><i><font color="Green">E9</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_2_1_1_2"><b>case </b></a><a NAME="E1:3_1_2_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:3_1_2_1_1_2"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/><div><i><font color="Green">E15</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:3_1_2_1_1_2_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font> = 
 <a href="cqc_lang.html#K2">IFEQ</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,0,0,<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>3</sub></font>)
<b>by </b><i><a class="txt" href="ami_4.html#E4:3_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="ami_4.html#E3:3_1_2_1_1_2"><i><font color="Green">E14</font></i></a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="ami_4.html#E1:3_1_2_1_1_2_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
;<br/><b>hence </b><a NAME="E3:3_1_2_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="ami_4.html#E1:3_1_2"><i><font color="Green">E13</font></i></a></i>;<br/></div><b>end;</b></div><font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font> = 
 <a href="cqc_lang.html#K2">IFEQ</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,0,0,<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>3</sub></font>)
<b>by </b><i><a class="txt" href="ami_4.html#E4:3_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="ami_4.html#E3:3_1_2_1_1_2"><i><font color="Green">E14</font></i></a></i>
<br/>.= 
<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>3</sub></font>)
<b>by </b><i><a class="txt" href="ami_4.html#E4:3_1_2_1_1_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
;<br/><b>hence </b><a NAME="E6:3_1_2_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>2</sub></font>)
 <b>by </b><i><a class="txt" href="ami_4.html#E5:3"><i><font color="Green">E7</font></i></a>, <a class="txt" href="ami_4.html#E1:3_1_2"><i><font color="Green">E13</font></i></a>, <a class="txt" href="ami_4.html#E3:3_1_2_1_1_2"><i><font color="Green">E14</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:3_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>2</sub></font>)
 ;<br/>


</div><b>end;</b></div>
<a NAME="E8:3_1"/><i><font color="Green">E13</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 2) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="nat_1.html#K4">mod</a> <font color="Maroon">H<sub>2</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="3_1_3"><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/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:3_1_3_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_3_1_1_1"><b>case </b></a><a NAME="E1:3_1_3_1_1_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_3_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:3_1_3_1_1_1_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>2</sub></font>) <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:3_1_3_1_1_1_1"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/><font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
 <a href="cqc_lang.html#K2">IFEQ</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,0,0,<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>2</sub></font>)
<b>by </b><i><a class="txt" href="ami_4.html#E4:3_1"><i><font color="Green">E10</font></i></a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="ami_4.html#E1:3_1_3_1_1_1_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="ami_4.html#E2:3_1_3_1_1_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
;<br/><b>hence </b><a NAME="E4:3_1_3_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="ami_4.html#E1:3_1_3_1_1_1"><i><font color="Green">E14</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E3:3_1_3_1_1_1"/><b>then </b><i><font color="Green">E15</font></i>: 
<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>2</sub></font>) <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><i><font color="Green">E16</font></i>: <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
<font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>

<br/>.= 
 <a href="cqc_lang.html#K2">IFEQ</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,0,0,<font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>)
<b>by </b><i><a class="txt" href="ami_4.html#E4:3_1"><i><font color="Green">E10</font></i></a></i>
<br/>.= 
<font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>)
<b>by </b><i><a class="txt" href="ami_4.html#E1:3_1_3_1_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
<br/>.= 
<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>2</sub></font>) <a href="nat_1.html#K4">mod</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>2</sub></font>)
<b>by </b><i><a class="txt" href="ami_4.html#E5:3"><i><font color="Green">E7</font></i></a>, <a class="txt" href="ami_4.html#E3:3_1_3_1_1_1"><i><font color="Green">E15</font></i></a></i>
;<br/><div><i><font color="Green">E17</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_3_1_1_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:3_1_3_1_1_1_3"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
 <a href="cqc_lang.html#K2">IFEQ</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,0,0,<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>2</sub></font>)
<b>by </b><i><a class="txt" href="ami_4.html#E4:3_1"><i><font color="Green">E10</font></i></a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="ami_4.html#E1:3_1_3_1_1_1_3"><i><font color="Green">E18</font></i></a>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
;<br/><b>hence </b><a NAME="E3:3_1_3_1_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="ami_4.html#E1:3_1_3_1_1_1"><i><font color="Green">E14</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E6:3_1_3_1_1_1"/><b>then </b><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
 <a href="cqc_lang.html#K2">IFEQ</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,0,0,<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>2</sub></font>)
<b>by </b><i><a class="txt" href="ami_4.html#E4:3_1"><i><font color="Green">E10</font></i></a></i>
<br/>.= 
<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>2</sub></font>)
<b>by </b><i><a class="txt" href="ami_4.html#E5:3_1_3_1_1_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
;<br/><b>hence </b><a NAME="E8:3_1_3_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="ami_4.html#E7:3_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="ami_4.html#E4:3_1_3_1_1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="ami_4.html#E6:3_1_3_1_1_1"><i><font color="Green">E18</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_3_1_1_2"><b>case </b></a><a NAME="E1:3_1_3_1_1_2"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>thus </b><font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span> = 
<font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>

<br/>.= 
 <a href="cqc_lang.html#K2">IFEQ</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,0,0,<font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>)
<b>by </b><i><a class="txt" href="ami_4.html#E4:3_1"><i><font color="Green">E10</font></i></a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="ami_4.html#E1:3_1_3_1_1_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="ami_4.html#E1:3_1_3_1_1_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:3_1_3"/>
<font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E10:3_1"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>2</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="nat_1.html#K8">hcf</a> <font color="Maroon">F<sub>4</sub></font>() &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1) <a href="hidden.html#R1">=</a> 0 )
 <b>from </b><i><a class="ref" href="nat_1.html#S8">NAT_1:sch 8</a>(<a class="txt" href="ami_4.html#E1:3_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="ami_4.html#E6:3_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="ami_4.html#E8:3_1"><i><font color="Green">E13</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon">c<sub>2</sub></font>
;<br/>

<a NAME="E11:3_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">F<sub>3</sub></font>() <a href="nat_1.html#K8">hcf</a> <font color="Maroon">F<sub>4</sub></font>() <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="ami_4.html#E1:3"><i><font color="Green">E3</font></i></a>, <a class="ref" href="newton.html#T71">NEWTON:71</a></i>;<br/>
<b>hence </b><a NAME="E12:3_1"/>
<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>2</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="nat_1.html#K8">hcf</a> <font color="Maroon">F<sub>4</sub></font>()
 <b>by </b><i><a class="txt" href="ami_4.html#E7:3_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="ami_4.html#E10:3_1"><i><font color="Green">E14</font></i></a></i>;<br/>

<b>thus </b><font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>2</sub></font>) = 
 <a href="cqc_lang.html#K2">IFEQ</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,0,0,<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>2</sub></font>)
<b>by </b><i><a class="txt" href="ami_4.html#E10:3_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="ami_4.html#E11:3_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="ami_4.html#E4:3_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="ami_4.html#E10:3_1"><i><font color="Green">E14</font></i></a></i>
;<br/>


</div>
