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

<b>set </b><font color="Maroon">c<sub>1</sub></font> = <font color="Maroon">F<sub>1</sub></font>();<br/>
<b>set </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">F<sub>2</sub></font>();<br/>
<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<b>assume </b><a NAME="E1:32_1"/><i><font color="Green">E26</font></i>: 
 <a href="msualg_6.html#K3" target="_self">TranslationRel</a> <font color="Maroon">F<sub>1</sub></font>() <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="msualg_6.html#M2" target="_self">Translation</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/>

<b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="rewrite1.html#M1">RedSequence</a> of  <a href="msualg_6.html#K3" target="_self">TranslationRel</a> <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">c<sub>7</sub></font> being  <a href="funcop_1.html#V1">Function-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E3:32_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="funct_7.html#K4">compose</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E4:32_1"/><i><font color="Green">E28</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></font></span>)</span> )
 <b>and </b><br/><a NAME="E5:32_1"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="funct_1.html#NM1">Function</a><br/>for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>() st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</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> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="msualg_6.html#R1" target="_self">is_e.translation_of</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="msualg_6.html#E1:32_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="msualg_6.html#D6" target="_self">Def6</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <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">c<sub>7</sub></font> implies ex <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>()ex <font color="Olive">b<sub>2</sub></font> being   <a href="msualg_6.html#M2" target="_self">Translation</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>ex <font color="Olive">b<sub>3</sub></font> being <a href="funcop_1.html#V1">Function-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp;  <a href="msualg_6.html#K3" target="_self">TranslationRel</a> <font color="Maroon">F<sub>1</sub></font>() <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="funct_7.html#K4">compose</a> <font color="Olive">b<sub>3</sub></font>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] ) );<br/>
<a NAME="E6:32_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E7:32_1"/><i><font color="Green">E31</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="32_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:32_1_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> implies ex <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>()ex <font color="Olive">b<sub>2</sub></font> being   <a href="msualg_6.html#M2" target="_self">Translation</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>ex <font color="Olive">b<sub>3</sub></font> being <a href="funcop_1.html#V1">Function-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp;  <a href="msualg_6.html#K3" target="_self">TranslationRel</a> <font color="Maroon">F<sub>1</sub></font>() <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="funct_7.html#K4">compose</a> <font color="Olive">b<sub>3</sub></font>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] ) )
 <b>and </b><br/><a NAME="E2:32_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>8</sub></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">c<sub>7</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="funct_1.html#NM1">Function</a> ;<br/>
<a NAME="E4:32_1_1"/>
( <font color="Maroon">c<sub>8</sub></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">c<sub>6</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <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">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="msualg_6.html#E4:32_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="msualg_6.html#E2:32_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_7.html#T24">FUNCT_7:24</a></i>;<br/>
<a NAME="E5:32_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <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>  <a href="msualg_6.html#K3" target="_self">TranslationRel</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="ref" href="rewrite1.html#D2">REWRITE1:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E7:32_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="msualg_6.html#R1" target="_self">is_e.translation_of</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="msualg_6.html#E5:32_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="msualg_6.html#E2:32_1_1"><i><font color="Green">E33</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>9</sub></font> as    <a href="msualg_6.html#M2" target="_self">Translation</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> <b>by </b><i><a class="ref" href="msualg_6.html#T17" target="_self">Th17</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_1_1_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>8</sub></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="32_1_1_1_1"><b>suppose </b></a><a NAME="E1:32_1_1_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> 1</span>)</span> as  <a href="funcop_1.html#V1">Function-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a> <b>by </b><i><a class="ref" href="finseq_1.html#T19">FINSEQ_1:19</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>12</sub></font> as    <a href="msualg_6.html#M2" target="_self">Translation</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>11</sub></font> <b>by </b><i><a class="txt" href="msualg_6.html#E4:32_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="msualg_6.html#E1:32_1_1_1_1"><i><font color="Green">E35</font></i></a></i>;<br/><a NAME="E4:32_1_1_1_1"/><i><font color="Green">E36</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>11</sub></font>
;<br/><b>take </b>
<font color="Maroon">c<sub>14</sub></font>
;<br/><b>take </b>
<font color="Maroon">c<sub>13</sub></font>
;<br/><b>thus </b><a NAME="E5:32_1_1_1_1"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="msualg_6.html#E1:32_1_1_1_1"><i><font color="Green">E35</font></i></a></i>;<br/><b>thus </b><a NAME="E6:32_1_1_1_1"/>
 <a href="msualg_6.html#K3" target="_self">TranslationRel</a> <font color="Maroon">F<sub>1</sub></font>() <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="msualg_6.html#E4:32_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="msualg_6.html#E7:32_1_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="msualg_6.html#E1:32_1_1_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="msualg_6.html#T17" target="_self">Th17</a></i>;<br/><a NAME="E7:32_1_1_1_1"/>
( 0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> &amp; 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1 )
 <b>by </b><i><a class="txt" href="msualg_6.html#E2:32_1_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="msualg_6.html#E1:32_1_1_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E8:32_1_1_1_1"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="msualg_6.html#E1:32_1_1_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="finseq_1.html#T21">FINSEQ_1:21</a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><a NAME="E9:32_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><b>hence </b><a NAME="E10:32_1_1_1_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="funct_7.html#K4">compose</a> <font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="msualg_6.html#E4:32_1_1_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="funct_7.html#T48">FUNCT_7:48</a></i>;<br/><a NAME="E11:32_1_1_1_1"/>
(  <a href="msualg_6.html#K3" target="_self">TranslationRel</a> <font color="Maroon">F<sub>1</sub></font>() <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp;  <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> is    <a href="msualg_6.html#M2" target="_self">Translation</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font>] )
 <b>by </b><i><a class="txt" href="msualg_6.html#E1:32"><i><font color="Green">E24</font></i></a>, <a class="ref" href="msualg_6.html#T16" target="_self">Th16</a>, <a class="ref" href="funct_2.html#T23">FUNCT_2:23</a>, <a class="ref" href="rewrite1.html#T13">REWRITE1:13</a></i>;<br/><b>hence </b><a NAME="E12:32_1_1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>11</sub></font>]
 <b>by </b><i><a class="txt" href="msualg_6.html#E2:32"><i><font color="Green">E25</font></i></a>, <a class="txt" href="msualg_6.html#E4:32_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="msualg_6.html#E7:32_1_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="msualg_6.html#E1:32_1_1_1_1"><i><font color="Green">E35</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2"><b>suppose </b></a><a NAME="E1:32_1_1_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:32_1_1_1_2"/><b>then </b><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="msualg_6.html#E2:32_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:32_1_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">c<sub>14</sub></font> being    <a href="msualg_6.html#M2" target="_self">Translation</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>15</sub></font> being  <a href="funcop_1.html#V1">Function-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E5:32_1_1_1_2"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp;  <a href="msualg_6.html#K3" target="_self">TranslationRel</a> <font color="Maroon">F<sub>1</sub></font>() <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="funct_7.html#K4">compose</a> <font color="Maroon">c<sub>15</sub></font>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>13</sub></font>] )
 <b>by </b><i><a class="txt" href="msualg_6.html#E1:32_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>11</sub></font>
;<br/><b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>12</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>14</sub></font> as    <a href="msualg_6.html#M2" target="_self">Translation</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>11</sub></font> <b>by </b><i><a class="txt" href="msualg_6.html#E7:32_1_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="msualg_6.html#E5:32_1_1_1_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="msualg_6.html#T19" target="_self">Th19</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>16</sub></font>
;<br/><b>take </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>15</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span>;<br/><b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> as   <a href="finseq_1.html#NM1">FinSequence</a> <b>by </b><i><a class="ref" href="finseq_1.html#T19">FINSEQ_1:19</a></i>;<br/><a NAME="E8:32_1_1_1_2"/>
(  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="msualg_6.html#E2:32_1_1_1_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><a NAME="E9:32_1_1_1_2"/><b>then </b><i><font color="Green">E37</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="msualg_6.html#E5:32_1_1_1_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_1.html#T21">FINSEQ_1:21</a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><a NAME="E10:32_1_1_1_2"/><b>then </b><i><font color="Green">E38</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>19</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:32_1_1_1_2_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><a NAME="E2:32_1_1_1_2_1"/><b>then </b>
( <font color="Maroon">c<sub>19</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>19</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E3:32_1_1_1_2_1"/><b>then </b>
( ( <font color="Maroon">c<sub>19</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 ) or <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T26">NAT_1:26</a></i>;<br/><a NAME="E4:32_1_1_1_2_1"/><b>then </b>
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>15</sub></font> or <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="msualg_6.html#E9:32_1_1_1_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:32_1_1_1_2_1"/><b>then </b>
( ( <font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> ) or <font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> )
 <b>by </b><i><a class="txt" href="msualg_6.html#E5:32_1_1_1_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="msualg_6.html#E9:32_1_1_1_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><b>hence </b><a NAME="E6:32_1_1_1_2_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="msualg_6.html#E10:32_1_1_1_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="msualg_6.html#E1:32_1_1_1_2_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E12:32_1_1_1_2"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="msualg_6.html#E10:32_1_1_1_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a></i>;<br/><b>thus </b><a NAME="E13:32_1_1_1_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><a NAME="E14:32_1_1_1_2"/>
 <a href="msualg_6.html#K3" target="_self">TranslationRel</a> <font color="Maroon">F<sub>1</sub></font>() <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="msualg_6.html#E7:32_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="msualg_6.html#T17" target="_self">Th17</a></i>;<br/><b>hence </b><a NAME="E15:32_1_1_1_2"/>
 <a href="msualg_6.html#K3" target="_self">TranslationRel</a> <font color="Maroon">F<sub>1</sub></font>() <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="msualg_6.html#E5:32_1_1_1_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="rewrite1.html#T17">REWRITE1:17</a></i>;<br/><b>thus </b><a NAME="E16:32_1_1_1_2"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a>  <a href="funct_7.html#K4">compose</a> <font color="Maroon">c<sub>17</sub></font>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="msualg_6.html#E5:32_1_1_1_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="funct_7.html#T43">FUNCT_7:43</a></i>;<br/><b>thus </b><a NAME="E17:32_1_1_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>11</sub></font>]
 <b>by </b><i><a class="txt" href="msualg_6.html#E2:32"><i><font color="Green">E25</font></i></a>, <a class="txt" href="msualg_6.html#E7:32_1_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="msualg_6.html#E5:32_1_1_1_2"><i><font color="Green">E36</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E8:32_1"/><i><font color="Green">E32</font></i>: 
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="msualg_6.html#E6:32_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="msualg_6.html#E7:32_1"><i><font color="Green">E31</font></i></a>);<br/></i>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_1_2"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_2_1"><b>suppose </b></a><a NAME="E1:32_1_2_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:32_1_2_1"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="txt" href="msualg_6.html#E3:32_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a>, <a class="ref" href="funct_7.html#T41">FUNCT_7:41</a></i>;<br/><b>hence </b><a NAME="E3:32_1_2_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>]
 <b>by </b><i><a class="txt" href="msualg_6.html#E1:32"><i><font color="Green">E24</font></i></a>, <a class="txt" href="msualg_6.html#E4:32_1"><i><font color="Green">E28</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_2_2"><b>suppose </b></a><a NAME="E1:32_1_2_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:32_1_2_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E3:32_1_2_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/><a NAME="E4:32_1_2_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:32_1_2_2"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a>, <a class="ref" href="relat_1.html#T98">RELAT_1:98</a></i>;<br/><a NAME="E6:32_1_2_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>()ex <font color="Olive">b<sub>2</sub></font> being   <a href="msualg_6.html#M2" target="_self">Translation</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>ex <font color="Olive">b<sub>3</sub></font> being <a href="funcop_1.html#V1">Function-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp;  <a href="msualg_6.html#K3" target="_self">TranslationRel</a> <font color="Maroon">F<sub>1</sub></font>() <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="funct_7.html#K4">compose</a> <font color="Olive">b<sub>3</sub></font>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] )
 <b>by </b><i><a class="txt" href="msualg_6.html#E8:32_1"><i><font color="Green">E32</font></i></a></i>;<br/><b>hence </b><a NAME="E7:32_1_2_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>]
 <b>by </b><i><a class="txt" href="msualg_6.html#E3:32_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="msualg_6.html#E4:32_1"><i><font color="Green">E28</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
