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<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon">A</font> be  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">s</font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>;<br/>





<b>thus </b><a NAME="E1:27"/>
 <a href="msualg_6.html#K3" target="_self">TranslationRel</a> <font color="Maroon">S</font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">s</font>,<font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="rewrite1.html#T13">REWRITE1:13</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_6.html#D6" target="_self">MSUALG_6:def 6</a><br/>

<a NAME="E2:27"/><i><font color="Green">E32</font></i>: 
 <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">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="funct_7.html#K4">compose</a> <a href="xboole_0.html#K1">{}</a> ,<span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_7.html#T41">FUNCT_7:41</a></i>;<br/>
<a NAME="E3:27"/><i><font color="Green">E34</font></i>: 
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s</font></span><a href="finseq_1.html#K12">*&gt;</a></span> is    <a href="rewrite1.html#M1">RedSequence</a> of  <a href="msualg_6.html#K3" target="_self">TranslationRel</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="rewrite1.html#T7">REWRITE1:7</a></i>;<br/>
<a NAME="E4:27"/><i><font color="Green">E43</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> 0 <a href="nat_1.html#K1">+</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> 0 &amp; <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>
<a NAME="E5:27"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">f</font> being  <a href="funct_1.html#NM1">Function</a><br/> for <font color="Olive">s1</font>, <font color="Olive">s2</font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <a href="xboole_0.html#K1">{}</a>  <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">s1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">s2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/><font color="Olive">f</font> <a href="msualg_6.html#R1" target="_self">is_e.translation_of</a> <font color="Maroon">A</font>,<font color="Olive">s1</font>,<font color="Olive">s2</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/>
<b>hence </b><a NAME="E6:27"/>
 ex <font color="Olive">q</font> being   <a href="rewrite1.html#M1">RedSequence</a> of  <a href="msualg_6.html#K3" target="_self">TranslationRel</a> <font color="Maroon">S</font> ex <font color="Olive">p</font> being <a href="funcop_1.html#V1">Function-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a> st <br/>(  <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">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="funct_7.html#K4">compose</a> <font color="Olive">p</font>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">q</font></span>)</span> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">f</font> being  <a href="funct_1.html#NM1">Function</a><br/> for <font color="Olive">s1</font>, <font color="Olive">s2</font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">p</font> &amp; <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">s1</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">s2</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/><font color="Olive">f</font> <a href="msualg_6.html#R1" target="_self">is_e.translation_of</a> <font color="Maroon">A</font>,<font color="Olive">s1</font>,<font color="Olive">s2</font> ) )
 <b>by </b><i>, , </i>;<br/>


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