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

<b>set </b><font color="Maroon" title="c1">S</font> = <font color="Maroon">F<sub>1</sub></font>();<br/>
<b>set </b><font color="Maroon" title="c2">A</font> = <font color="Maroon">F<sub>2</sub></font>();<br/>
<b>let </b><font color="Maroon" title="c3">s1</font>, <font color="Maroon" title="c4">s2</font> be   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>(); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="msualg_6.html#K3" title="MSUALG_6:func.3">TranslationRel</a> <font color="Maroon">F<sub>1</sub></font>() <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c4">s2</font> implies for <font color="Olive" title="b1">t</font> being   <a href="msualg_6.html#M2" title="MSUALG_6:mode.2">Translation</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c4">s2</font> holds  <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">t</font>,<font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c4">s2</font>] )</span><br/>

<b>assume </b><a NAME="E1:32_1"/><i><font color="Green" title="E26">A3</font></i>: 
 <a href="msualg_6.html#K3" title="MSUALG_6:func.3">TranslationRel</a> <font color="Maroon">F<sub>1</sub></font>() <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c4">s2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">t</font> being   <a href="msualg_6.html#M2" title="MSUALG_6:mode.2">Translation</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c4">s2</font> holds  <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">t</font>,<font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c4">s2</font>]</span><br/>

<b>let </b><font color="Maroon" title="c5">t</font> be    <a href="msualg_6.html#M2" title="MSUALG_6:mode.2">Translation</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c4">s2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">t</font>,<font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c4">s2</font>]</span><br/>

<b>consider </b><font color="Maroon" title="c6">q</font> being    <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of  <a href="msualg_6.html#K3" title="MSUALG_6:func.3">TranslationRel</a> <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon" title="c7">p</font> being  <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a> <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E3:32_1"/><i><font color="Green" title="E27">A4</font></i>: 
<font color="Maroon" title="c5">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <font color="Maroon" title="c7">p</font>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">s1</font></span>)</span>
 <b>and </b><br/><a NAME="E4:32_1"/><i><font color="Green" title="E28">A5</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; <font color="Maroon" title="c4">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">q</font></span>)</span> )
 <b>and </b><br/><a NAME="E5:32_1"/><i><font color="Green" title="E29">A6</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><br/> for <font color="Olive" title="b3">s1</font>, <font color="Olive" title="b4">s2</font> being  <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</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>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">p</font> &amp; <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b4">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> holds <br/><font color="Olive" title="b2">f</font> <a href="msualg_6.html#R1" title="MSUALG_6:pred.1">is_e.translation_of</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive" title="b3">s1</font>,<font color="Olive" title="b4">s2</font>
 <b>by </b><i><a class="txt" href="#E1:32_1"><i><font color="Green" title="E26">A3</font></i></a>, <a class="ref" href="msualg_6.html#D6" target="_self" title="MSUALG_6:def.6">Def6</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">p</font> implies  ex <font color="Olive" title="b1">s</font> being  <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>() ex <font color="Olive" title="b2">t</font> being   <a href="msualg_6.html#M2" title="MSUALG_6:mode.2">Translation</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c3">s1</font>,<font color="Olive" title="b1">s</font> ex <font color="Olive" title="b3">p'</font> being <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a> <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>( <font color="Olive" title="b3">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> $1</span>)</span> &amp; <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">$1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp;  <a href="msualg_6.html#K3" title="MSUALG_6:func.3">TranslationRel</a> <font color="Maroon">F<sub>1</sub></font>() <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c3">s1</font>,<font color="Olive" title="b1">s</font> &amp; <font color="Olive" title="b2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <font color="Olive" title="b3">p'</font>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">s1</font></span>)</span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">t</font>,<font color="Maroon" title="c3">s1</font>,<font color="Olive" title="b1">s</font>] ) );<br/>
<a NAME="E6:32_1"/><i><font color="Green" title="E30">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="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E7:32_1"/><i><font color="Green" title="E31">A8</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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" title="c8">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">i</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:32_1_1"/><i><font color="Green" title="E32">A9</font></i>: 
( <font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">p</font> implies  ex <font color="Olive" title="b1">s</font> being  <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>() ex <font color="Olive" title="b2">t</font> being   <a href="msualg_6.html#M2" title="MSUALG_6:mode.2">Translation</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c3">s1</font>,<font color="Olive" title="b1">s</font> ex <font color="Olive" title="b3">p'</font> being <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a> <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>( <font color="Olive" title="b3">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c8">i</font></span>)</span> &amp; <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp;  <a href="msualg_6.html#K3" title="MSUALG_6:func.3">TranslationRel</a> <font color="Maroon">F<sub>1</sub></font>() <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c3">s1</font>,<font color="Olive" title="b1">s</font> &amp; <font color="Olive" title="b2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <font color="Olive" title="b3">p'</font>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">s1</font></span>)</span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">t</font>,<font color="Maroon" title="c3">s1</font>,<font color="Olive" title="b1">s</font>] ) )
 <b>and </b><br/><a NAME="E2:32_1_1"/><i><font color="Green" title="E33">A10</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">s</font> being  <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>() ex <font color="Olive" title="b2">t</font> being   <a href="msualg_6.html#M2" title="MSUALG_6:mode.2">Translation</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c3">s1</font>,<font color="Olive" title="b1">s</font> ex <font color="Olive" title="b3">p'</font> being <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a> <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>( <font color="Olive" title="b3">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp;  <a href="msualg_6.html#K3" title="MSUALG_6:func.3">TranslationRel</a> <font color="Maroon">F<sub>1</sub></font>() <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c3">s1</font>,<font color="Olive" title="b1">s</font> &amp; <font color="Olive" title="b2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <font color="Olive" title="b3">p'</font>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">s1</font></span>)</span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">t</font>,<font color="Maroon" title="c3">s1</font>,<font color="Olive" title="b1">s</font>] )</span><br/>

<b>reconsider </b><font color="Maroon" title="c9">f</font> = <font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/>
<a NAME="E4:32_1_1"/>
( <font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">q</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">q</font> )
 
<b>by </b><i><a class="txt" href="#E4:32_1"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="#E2:32_1_1"><i><font color="Green" title="E33">A10</font></i></a>, <a class="ref" href="funct_7.html#T24" title="FUNCT_7:th.24">FUNCT_7:24</a></i>;<br/>
<a NAME="E5:32_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_6.html#K3" title="MSUALG_6:func.3">TranslationRel</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="ref" href="rewrite1.html#D2" title="REWRITE1:def.2">REWRITE1:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">v1</font> = <font color="Maroon" title="c6">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>, <font color="Maroon" title="c11">v2</font> = <font color="Maroon" title="c6">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E7:32_1_1"/><i><font color="Green" title="E34">A11</font></i>: 
<font color="Maroon" title="c9">f</font> <a href="msualg_6.html#R1" title="MSUALG_6:pred.1">is_e.translation_of</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c11">v2</font>
 
<b>by </b><i><a class="txt" href="#E5:32_1"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="#E2:32_1_1"><i><font color="Green" title="E33">A10</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">t</font> = <font color="Maroon" title="c9">f</font> as    <a href="msualg_6.html#M2" title="MSUALG_6:mode.2">Translation</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c11">v2</font> <b>by </b><i><a class="ref" href="msualg_6.html#T17" target="_self" title="MSUALG_6:th.17">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" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<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" title="E35">A12</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">s</font> being  <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>() ex <font color="Olive" title="b2">t</font> being   <a href="msualg_6.html#M2" title="MSUALG_6:mode.2">Translation</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c3">s1</font>,<font color="Olive" title="b1">s</font> ex <font color="Olive" title="b3">p'</font> being <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a> <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>( <font color="Olive" title="b3">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp;  <a href="msualg_6.html#K3" title="MSUALG_6:func.3">TranslationRel</a> <font color="Maroon">F<sub>1</sub></font>() <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c3">s1</font>,<font color="Olive" title="b1">s</font> &amp; <font color="Olive" title="b2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <font color="Olive" title="b3">p'</font>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">s1</font></span>)</span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">t</font>,<font color="Maroon" title="c3">s1</font>,<font color="Olive" title="b1">s</font>] )</span><br/><div class="add"><b>reconsider </b><font color="Maroon" title="c13">p'</font> = <font color="Maroon" title="c7">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1</span>)</span> as  <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a> <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> <b>by </b><i><a class="ref" href="finseq_1.html#T19" title="FINSEQ_1:th.19">FINSEQ_1:19</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c14">t</font> = <font color="Maroon" title="c12">t</font> as    <a href="msualg_6.html#M2" title="MSUALG_6:mode.2">Translation</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c11">v2</font> <b>by </b><i><a class="txt" href="#E4:32_1"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="#E1:32_1_1_1_1"><i><font color="Green" title="E35">A12</font></i></a></i>;<br/><a NAME="E4:32_1_1_1_1"/><i><font color="Green" title="E36">A13</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c14">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">s1</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><b>take </b>
<font color="Maroon" title="c11">v2</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">t</font> being   <a href="msualg_6.html#M2" title="MSUALG_6:mode.2">Translation</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c11">v2</font> ex <font color="Olive" title="b2">p'</font> being <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a> <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>( <font color="Olive" title="b2">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c11">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp;  <a href="msualg_6.html#K3" title="MSUALG_6:func.3">TranslationRel</a> <font color="Maroon">F<sub>1</sub></font>() <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c11">v2</font> &amp; <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <font color="Olive" title="b2">p'</font>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">s1</font></span>)</span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">t</font>,<font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c11">v2</font>] )</span><br/><b>take </b>
<font color="Maroon" title="c14">t</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">p'</font> being <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a> <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>( <font color="Olive" title="b1">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c11">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp;  <a href="msualg_6.html#K3" title="MSUALG_6:func.3">TranslationRel</a> <font color="Maroon">F<sub>1</sub></font>() <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c11">v2</font> &amp; <font color="Maroon" title="c14">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <font color="Olive" title="b1">p'</font>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">s1</font></span>)</span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c14">t</font>,<font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c11">v2</font>] )</span><br/><b>take </b>
<font color="Maroon" title="c13">p'</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c13">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c11">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp;  <a href="msualg_6.html#K3" title="MSUALG_6:func.3">TranslationRel</a> <font color="Maroon">F<sub>1</sub></font>() <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c11">v2</font> &amp; <font color="Maroon" title="c14">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <font color="Maroon" title="c13">p'</font>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">s1</font></span>)</span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c14">t</font>,<font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c11">v2</font>] )</span><br/><b>thus </b><a NAME="E5:32_1_1_1_1"/>
( <font color="Maroon" title="c13">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c11">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="#E1:32_1_1_1_1"><i><font color="Green" title="E35">A12</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="msualg_6.html#K3" title="MSUALG_6:func.3">TranslationRel</a> <font color="Maroon">F<sub>1</sub></font>() <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c11">v2</font> &amp; <font color="Maroon" title="c14">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <font color="Maroon" title="c13">p'</font>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">s1</font></span>)</span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c14">t</font>,<font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c11">v2</font>] )</span><br/><b>thus </b><a NAME="E6:32_1_1_1_1"/>
 <a href="msualg_6.html#K3" title="MSUALG_6:func.3">TranslationRel</a> <font color="Maroon">F<sub>1</sub></font>() <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c11">v2</font>
 <b>by </b><i><a class="txt" href="#E4:32_1"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="#E7:32_1_1"><i><font color="Green" title="E34">A11</font></i></a>, <a class="txt" href="#E1:32_1_1_1_1"><i><font color="Green" title="E35">A12</font></i></a>, <a class="ref" href="msualg_6.html#T17" target="_self" title="MSUALG_6:th.17">Th17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c14">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <font color="Maroon" title="c13">p'</font>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">s1</font></span>)</span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c14">t</font>,<font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c11">v2</font>] )</span><br/><a NAME="E7:32_1_1_1_1"/>
( <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p</font> &amp; 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1 )
 <b>by </b><i><a class="txt" href="#E2:32_1_1"><i><font color="Green" title="E33">A10</font></i></a>, <a class="txt" href="#E1:32_1_1_1_1"><i><font color="Green" title="E35">A12</font></i></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E8:32_1_1_1_1"/><b>then </b>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c13">p'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">t</font> )
 <b>by </b><i><a class="txt" href="#E1:32_1_1_1_1"><i><font color="Green" title="E35">A12</font></i></a>, <a class="ref" href="finseq_1.html#T21" title="FINSEQ_1:th.21">FINSEQ_1:21</a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/><a NAME="E9:32_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c13">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c14">t</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/><b>hence </b><a NAME="E10:32_1_1_1_1"/>
<font color="Maroon" title="c14">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <font color="Maroon" title="c13">p'</font>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">s1</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:32_1_1_1_1"><i><font color="Green" title="E36">A13</font></i></a>, <a class="ref" href="funct_7.html#T48" title="FUNCT_7:th.48">FUNCT_7:48</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c14">t</font>,<font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c11">v2</font>]</span><br/><a NAME="E11:32_1_1_1_1"/>
(  <a href="msualg_6.html#K3" title="MSUALG_6:func.3">TranslationRel</a> <font color="Maroon">F<sub>1</sub></font>() <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c3">s1</font> &amp;  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">s1</font></span>)</span> is    <a href="msualg_6.html#M2" title="MSUALG_6:mode.2">Translation</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c3">s1</font> &amp; <font color="Maroon" title="c14">t</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">s1</font></span>)</span></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c14">t</font> &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">s1</font></span>)</span>,<font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c3">s1</font>] )
 <b>by </b><i><a class="txt" href="#E1:32"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="msualg_6.html#T16" target="_self" title="MSUALG_6:th.16">Th16</a>, <a class="ref" href="funct_2.html#T23" title="FUNCT_2:th.23">FUNCT_2:23</a>, <a class="ref" href="rewrite1.html#T13" title="REWRITE1:th.13">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" title="c14">t</font>,<font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c11">v2</font>]
 <b>by </b><i><a class="txt" href="#E2:32"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="#E4:32_1"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="#E5:32_1"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="#E2:32_1_1"><i><font color="Green" title="E33">A10</font></i></a>, <a class="txt" href="#E1:32_1_1_1_1"><i><font color="Green" title="E35">A12</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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" title="c8">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">s</font> being  <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>() ex <font color="Olive" title="b2">t</font> being   <a href="msualg_6.html#M2" title="MSUALG_6:mode.2">Translation</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c3">s1</font>,<font color="Olive" title="b1">s</font> ex <font color="Olive" title="b3">p'</font> being <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a> <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>( <font color="Olive" title="b3">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp;  <a href="msualg_6.html#K3" title="MSUALG_6:func.3">TranslationRel</a> <font color="Maroon">F<sub>1</sub></font>() <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c3">s1</font>,<font color="Olive" title="b1">s</font> &amp; <font color="Olive" title="b2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <font color="Olive" title="b3">p'</font>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">s1</font></span>)</span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">t</font>,<font color="Maroon" title="c3">s1</font>,<font color="Olive" title="b1">s</font>] )</span><br/><div class="add"><a NAME="E2:32_1_1_1_2"/><b>then </b><i><font color="Green" title="E35">A14</font></i>: 
( <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p</font> &amp; <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>by </b><i><a class="txt" href="#E2:32_1_1"><i><font color="Green" title="E33">A10</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:32_1_1_1_2"/><b>then </b>
( <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p</font> &amp; <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>then consider </b><font color="Maroon" title="c13">s</font> being   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon" title="c14">t'</font> being    <a href="msualg_6.html#M2" title="MSUALG_6:mode.2">Translation</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c13">s</font>, <font color="Maroon" title="c15">p'</font> being  <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a> <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E5:32_1_1_1_2"/><i><font color="Green" title="E36">A15</font></i>: 
( <font color="Maroon" title="c15">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c8">i</font></span>)</span> &amp; <font color="Maroon" title="c13">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp;  <a href="msualg_6.html#K3" title="MSUALG_6:func.3">TranslationRel</a> <font color="Maroon">F<sub>1</sub></font>() <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c13">s</font> &amp; <font color="Maroon" title="c14">t'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <font color="Maroon" title="c15">p'</font>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">s1</font></span>)</span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c14">t'</font>,<font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c13">s</font>] )
 <b>by </b><i><a class="txt" href="#E1:32_1_1"><i><font color="Green" title="E32">A9</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>take </b>
<font color="Maroon" title="c11">v2</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">t</font> being   <a href="msualg_6.html#M2" title="MSUALG_6:mode.2">Translation</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c11">v2</font> ex <font color="Olive" title="b2">p'</font> being <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a> <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>( <font color="Olive" title="b2">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c11">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp;  <a href="msualg_6.html#K3" title="MSUALG_6:func.3">TranslationRel</a> <font color="Maroon">F<sub>1</sub></font>() <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c11">v2</font> &amp; <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <font color="Olive" title="b2">p'</font>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">s1</font></span>)</span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">t</font>,<font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c11">v2</font>] )</span><br/><b>reconsider </b><font color="Maroon" title="c16">T</font> = <font color="Maroon" title="c12">t</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c14">t'</font> as    <a href="msualg_6.html#M2" title="MSUALG_6:mode.2">Translation</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c11">v2</font> <b>by </b><i><a class="txt" href="#E5:32_1"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="#E2:32_1_1"><i><font color="Green" title="E33">A10</font></i></a>, <a class="txt" href="#E5:32_1_1_1_2"><i><font color="Green" title="E36">A15</font></i></a>, <a class="ref" href="msualg_6.html#T19" target="_self" title="MSUALG_6:th.19">Th19</a></i>;<br/><b>take </b>
<font color="Maroon" title="c16">T</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">p'</font> being <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a> <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>( <font color="Olive" title="b1">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c11">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp;  <a href="msualg_6.html#K3" title="MSUALG_6:func.3">TranslationRel</a> <font color="Maroon">F<sub>1</sub></font>() <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c11">v2</font> &amp; <font color="Maroon" title="c16">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <font color="Olive" title="b1">p'</font>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">s1</font></span>)</span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c16">T</font>,<font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c11">v2</font>] )</span><br/><b>take </b><font color="Maroon" title="c17">y</font> = <font color="Maroon" title="c15">p'</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c9">f</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c17">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c11">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp;  <a href="msualg_6.html#K3" title="MSUALG_6:func.3">TranslationRel</a> <font color="Maroon">F<sub>1</sub></font>() <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c11">v2</font> &amp; <font color="Maroon" title="c16">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <font color="Maroon" title="c17">y</font>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">s1</font></span>)</span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c16">T</font>,<font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c11">v2</font>] )</span><br/><b>reconsider </b><font color="Maroon" title="c18">pp</font> = <font color="Maroon" title="c7">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> <b>by </b><i><a class="ref" href="finseq_1.html#T19" title="FINSEQ_1:th.19">FINSEQ_1:19</a></i>;<br/><a NAME="E8:32_1_1_1_2"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c9">f</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p</font> &amp; <font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p</font> )
 <b>by </b><i><a class="txt" href="#E2:32_1_1_1_2"><i><font color="Green" title="E35">A14</font></i></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E9:32_1_1_1_2"/><b>then </b><i><font color="Green" title="E37">A16</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">p'</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">pp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>by </b><i><a class="txt" href="#E5:32_1_1_1_2"><i><font color="Green" title="E36">A15</font></i></a>, <a class="ref" href="finseq_1.html#T21" title="FINSEQ_1:th.21">FINSEQ_1:21</a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/><a NAME="E10:32_1_1_1_2"/><b>then </b><i><font color="Green" title="E38">A17</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c18">pp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c17">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">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" title="c19">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c19">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> implies <font color="Maroon" title="c17">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">pp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">k</font> )</span><br/><b>assume </b><a NAME="E1:32_1_1_1_2_1"/><i><font color="Green" title="E39">A18</font></i>: 
<font color="Maroon" title="c19">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c17">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">pp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">k</font></span><br/><a NAME="E2:32_1_1_1_2_1"/><b>then </b>
( <font color="Maroon" title="c19">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c19">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E3:32_1_1_1_2_1"/><b>then </b>
( ( <font color="Maroon" title="c19">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font> &amp; <font color="Maroon" title="c19">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 ) or <font color="Maroon" title="c19">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></i>;<br/><a NAME="E4:32_1_1_1_2_1"/><b>then </b>
( <font color="Maroon" title="c19">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">p'</font> or <font color="Maroon" title="c19">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>by </b><i><a class="txt" href="#E9:32_1_1_1_2"><i><font color="Green" title="E37">A16</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:32_1_1_1_2_1"/><b>then </b>
( ( <font color="Maroon" title="c17">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">p'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">k</font> &amp; <font color="Maroon" title="c15">p'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">k</font> ) or <font color="Maroon" title="c17">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">k</font> )
 <b>by </b><i><a class="txt" href="#E5:32_1_1_1_2"><i><font color="Green" title="E36">A15</font></i></a>, <a class="txt" href="#E9:32_1_1_1_2"><i><font color="Green" title="E37">A16</font></i></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/><b>hence </b><a NAME="E6:32_1_1_1_2_1"/>
<font color="Maroon" title="c17">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">pp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">k</font>
 <b>by </b><i><a class="txt" href="#E10:32_1_1_1_2"><i><font color="Green" title="E38">A17</font></i></a>, <a class="txt" href="#E1:32_1_1_1_2_1"><i><font color="Green" title="E39">A18</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E12:32_1_1_1_2"/>
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E10:32_1_1_1_2"><i><font color="Green" title="E38">A17</font></i></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c11">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp;  <a href="msualg_6.html#K3" title="MSUALG_6:func.3">TranslationRel</a> <font color="Maroon">F<sub>1</sub></font>() <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c11">v2</font> &amp; <font color="Maroon" title="c16">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <font color="Maroon" title="c17">y</font>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">s1</font></span>)</span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c16">T</font>,<font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c11">v2</font>] )</span><br/><b>thus </b><a NAME="E13:32_1_1_1_2"/>
<font color="Maroon" title="c11">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="msualg_6.html#K3" title="MSUALG_6:func.3">TranslationRel</a> <font color="Maroon">F<sub>1</sub></font>() <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c11">v2</font> &amp; <font color="Maroon" title="c16">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <font color="Maroon" title="c17">y</font>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">s1</font></span>)</span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c16">T</font>,<font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c11">v2</font>] )</span><br/><a NAME="E14:32_1_1_1_2"/>
 <a href="msualg_6.html#K3" title="MSUALG_6:func.3">TranslationRel</a> <font color="Maroon">F<sub>1</sub></font>() <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c11">v2</font>
 <b>by </b><i><a class="txt" href="#E7:32_1_1"><i><font color="Green" title="E34">A11</font></i></a>, <a class="ref" href="msualg_6.html#T17" target="_self" title="MSUALG_6:th.17">Th17</a></i>;<br/><b>hence </b><a NAME="E15:32_1_1_1_2"/>
 <a href="msualg_6.html#K3" title="MSUALG_6:func.3">TranslationRel</a> <font color="Maroon">F<sub>1</sub></font>() <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c11">v2</font>
 <b>by </b><i><a class="txt" href="#E5:32_1_1_1_2"><i><font color="Green" title="E36">A15</font></i></a>, <a class="ref" href="rewrite1.html#T17" title="REWRITE1:th.17">REWRITE1:17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c16">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <font color="Maroon" title="c17">y</font>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">s1</font></span>)</span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c16">T</font>,<font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c11">v2</font>] )</span><br/><b>thus </b><a NAME="E16:32_1_1_1_2"/>
<font color="Maroon" title="c16">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <font color="Maroon" title="c17">y</font>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">s1</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E5:32_1_1_1_2"><i><font color="Green" title="E36">A15</font></i></a>, <a class="ref" href="funct_7.html#T43" title="FUNCT_7:th.43">FUNCT_7:43</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c16">T</font>,<font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c11">v2</font>]</span><br/><b>thus </b><a NAME="E17:32_1_1_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c16">T</font>,<font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c11">v2</font>]
 <b>by </b><i><a class="txt" href="#E2:32"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="#E5:32_1"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="#E2:32_1_1"><i><font color="Green" title="E33">A10</font></i></a>, <a class="txt" href="#E5:32_1_1_1_2"><i><font color="Green" title="E36">A15</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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" title="E32">A19</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E6:32_1"><i><font color="Green" title="E30">A7</font></i></a>, <a class="txt" href="#E7:32_1"><i><font color="Green" title="E31">A8</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" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c7">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">t</font>,<font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c4">s2</font>]</span><br/><div class="add"><a NAME="E2:32_1_2_1"/><b>then </b>
( <font color="Maroon" title="c5">t</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">s1</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="#E3:32_1"><i><font color="Green" title="E27">A4</font></i></a>, <a class="ref" href="funct_7.html#T41" title="FUNCT_7:th.41">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" title="c5">t</font>,<font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c4">s2</font>]
 <b>by </b><i><a class="txt" href="#E1:32"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="#E4:32_1"><i><font color="Green" title="E28">A5</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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" title="c7">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">t</font>,<font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c4">s2</font>]</span><br/><div class="add"><a NAME="E2:32_1_2_2"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E3:32_1_2_2"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E4:32_1_2_2"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E5:32_1_2_2"/><b>then </b>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">p</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p</font></span>)</span> &amp; <font color="Maroon" title="c7">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="ref" href="relat_1.html#T98" title="RELAT_1:th.98">RELAT_1:98</a></i>;<br/><a NAME="E6:32_1_2_2"/><b>then </b>
 ex <font color="Olive" title="b1">s</font> being  <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>() ex <font color="Olive" title="b2">t</font> being   <a href="msualg_6.html#M2" title="MSUALG_6:mode.2">Translation</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c3">s1</font>,<font color="Olive" title="b1">s</font> ex <font color="Olive" title="b3">p'</font> being <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a> <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>( <font color="Olive" title="b3">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font> &amp; <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp;  <a href="msualg_6.html#K3" title="MSUALG_6:func.3">TranslationRel</a> <font color="Maroon">F<sub>1</sub></font>() <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c3">s1</font>,<font color="Olive" title="b1">s</font> &amp; <font color="Olive" title="b2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <font color="Olive" title="b3">p'</font>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">s1</font></span>)</span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">t</font>,<font color="Maroon" title="c3">s1</font>,<font color="Olive" title="b1">s</font>] )
 <b>by </b><i><a class="txt" href="#E8:32_1"><i><font color="Green" title="E32">A19</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" title="c5">t</font>,<font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c4">s2</font>]
 <b>by </b><i><a class="txt" href="#E3:32_1"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="#E4:32_1"><i><font color="Green" title="E28">A5</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
