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

<b>let </b><font color="Maroon">R</font> be   <a href="relat_1.html#NM1">Relation</a>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:74_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">R</font> is <a href="rewrite1.html#V7" target="_self">confluent</a>
 ;<br/><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">c</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E2:74_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">R</font> <a href="rewrite1.html#R1" target="_self">reduces</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 ;<br/><a NAME="E3:74_1"/><b>then </b>
<font color="Maroon">R</font> <a href="rewrite1.html#R1" target="_self">reduces</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>
 <b>by </b><i/>;<br/><a NAME="E4:74_1"/><b>then </b>
<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="rewrite1.html#R6" target="_self">are_divergent_wrt</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="rewrite1.html#D4" target="_self">Def4</a>, <a class="txt" href="rewrite1.html#E10:74"><i><font color="Green">E39</font></i></a></i>;<br/><b>hence </b><a NAME="E5:74_1"/>
<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="rewrite1.html#R5" target="_self">are_convergent_wrt</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="rewrite1.html#D3" target="_self">Def3</a>, <a class="ref" href="rewrite1.html#T9" target="_self">Th9</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:74"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Maroon">R</font> <a href="rewrite1.html#R1" target="_self">reduces</a> <font color="Olive">a</font>,<font color="Olive">b</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> holds <br/><font color="Olive">b</font>,<font color="Olive">c</font> <a href="rewrite1.html#R5" target="_self">are_convergent_wrt</a> <font color="Maroon">R</font>
 ;<br/>

<b>let </b><font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">c</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="rewrite1.html#D22" target="_self">REWRITE1:def 22</a><br/>

<b>given </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E3:74"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">R</font> <a href="rewrite1.html#R1" target="_self">reduces</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">R</font> <a href="rewrite1.html#R1" target="_self">reduces</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="rewrite1.html#D8" target="_self">REWRITE1:def 8</a><br/>

<b>consider </b><font color="Maroon">p</font> being    <a href="rewrite1.html#M1" target="_self">RedSequence</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E5:74"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="rewrite1.html#D4" target="_self">Def4</a>, </i>;<br/>
<a NAME="E6:74"/><i><font color="Green">E35</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/>
<b>consider </b><font color="Maroon">q</font> being    <a href="rewrite1.html#M1" target="_self">RedSequence</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E8:74"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> )
 <b>by </b><i><a class="ref" href="rewrite1.html#D4" target="_self">Def4</a>, </i>;<br/>
<a NAME="E9:74"/><i><font color="Green">E38</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<b>by </b><i>, <a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>2</sub></font> <a href="rewrite1.html#R5" target="_self">are_convergent_wrt</a> <font color="Maroon">R</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> implies for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font>,<font color="Olive">j</font>] );<br/>
<a NAME="E10:74"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<b>by </b><i/>;<br/>
<a NAME="E11:74"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:74_2"/>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> implies for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font>,<font color="Olive">j</font>] )
 <b>and </b><br/><a NAME="E2:74_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q</font> implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1,<font color="Maroon">a<sub>1</sub></font>] );<br/>
<a NAME="E3:74_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[0]
 
<b>by </b><i/>;<br/>
<a NAME="E4:74_2"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">j</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">j</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:74_2_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q</font> implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1,<font color="Maroon">j</font>] )
 <b>and </b><br/><a NAME="E2:74_2_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q</font>
 ;<br/>

<a NAME="E3:74_2_1"/>
( 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 )
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E4:74_2_1"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">R</font> <a href="rewrite1.html#R1" target="_self">reduces</a> <font color="Maroon">a</font>,<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="rewrite1.html#T12" target="_self">Th12</a>, , </i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:74_2_1_1"/>
( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span><b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_2_1_1_1"><b>suppose </b></a><a NAME="E1:74_2_1_1_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:74_2_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1,<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="ref" href="rewrite1.html#T14" target="_self">Th14</a>, <a class="ref" href="rewrite1.html#T21" target="_self">Th21</a>, <a class="txt" href="rewrite1.html#E2:74_2"><i><font color="Green">E43</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_2_1_1_2"><b>suppose </b></a><a NAME="E1:74_2_1_1_2"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:74_2_1_1_2"/><b>then </b><i><font color="Green">E49</font></i>: 
( <font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font> )
 <b>by </b><i><a class="ref" href="rewrite1.html#T20" target="_self">Th20</a>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:74_2_1_1_2"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q</font>
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">d</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:74_2_1_1_2"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">R</font> <a href="rewrite1.html#R1" target="_self">reduces</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">d</font> &amp; <font color="Maroon">R</font> <a href="rewrite1.html#R1" target="_self">reduces</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">d</font> )
 <b>by </b><i><a class="ref" href="rewrite1.html#T19" target="_self">Th19</a>, <a class="ref" href="rewrite1.html#T22" target="_self">Th22</a>, <a class="txt" href="rewrite1.html#E9:74"><i><font color="Green">E38</font></i></a>, </i>;<br/><a NAME="E6:74_2_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="rewrite1.html#T20" target="_self">Th20</a>, <a class="ref" href="rewrite1.html#T23" target="_self">Th23</a>, </i>;<br/><a NAME="E7:74_2_1_1_2"/><b>then </b>
<font color="Maroon">d</font>,<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="rewrite1.html#R5" target="_self">are_convergent_wrt</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="rewrite1.html#D3" target="_self">Def3</a>, </i>;<br/><b>hence </b><a NAME="E8:74_2_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1,<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i>, <a class="txt" href="rewrite1.html#E2:74_2_1"><i><font color="Green">E47</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>thus </b><a NAME="E5:74_2"/>
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">j</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="ref" href="rewrite1.html#T18" target="_self">Th18</a>, );</i><br/>


</div><b>end;</b></div>
<a NAME="E12:74"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="ref" href="rewrite1.html#T16" target="_self">Th16</a>, <a class="ref" href="rewrite1.html#T17" target="_self">Th17</a>);<br/></i>
<b>hence </b><a NAME="E13:74"/>
<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="rewrite1.html#R5" target="_self">are_convergent_wrt</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="rewrite1.html#T12" target="_self">Th12</a>, <a class="ref" href="rewrite1.html#T13" target="_self">Th13</a>, <a class="ref" href="rewrite1.html#T14" target="_self">Th14</a>, <a class="ref" href="rewrite1.html#T15" target="_self">Th15</a></i>;<br/>


</div>
