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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">s1</font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>, <font color="Maroon">s2</font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>







<b>assume </b><a NAME="E1:28"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">f</font> <a href="msualg_6.html#R1" target="_self">is_e.translation_of</a> <font color="Maroon">A</font>,<font color="Maroon">s1</font>,<font color="Maroon">s2</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">g</font> = <font color="Maroon">f</font> as   <a href="funct_2.html#NM1">Function</a> of 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">s1</font>,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">s2</font> <b>by </b><i/>;<br/>
<a NAME="E3:28"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> 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">s1</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:28"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="funct_7.html#K4">compose</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">f</font></span><a href="finseq_1.html#K9">*&gt;</a></span>,<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">s1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_7.html#T48">FUNCT_7:48</a></i>;<br/>
<a NAME="E5:28"/><i><font color="Green">E43</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">s1</font>,<font color="Maroon">s2</font></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">S</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:28"/><i><font color="Green">E46</font></i>: 
 <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">s1</font>,<font color="Maroon">s2</font>
 <b>by </b><i><a class="ref" href="rewrite1.html#T16">REWRITE1:16</a></i>;<br/>

<a NAME="E7:28"/><i><font color="Green">E55</font></i>: 
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">s1</font>,<font color="Maroon">s2</font></span><a href="finseq_4.html#K2">*&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#T8">REWRITE1:8</a></i>;<br/>
<a NAME="E8:28"/><i><font color="Green">E56</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">s1</font>,<font color="Maroon">s2</font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> 1 &amp;  <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">f</font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1 &amp; <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">s1</font>,<font color="Maroon">s2</font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> &amp; <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">s1</font>,<font color="Maroon">s2</font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> &amp; <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">f</font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">g</font> be   <a href="funct_1.html#NM1">Function</a>;<br/><b>let </b><font color="Maroon">w1</font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>, <font color="Maroon">w2</font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>;<br/><b>assume </b><a NAME="E1:28_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">f</font></span><a href="finseq_1.html#K9">*&gt;</a></span>
 ;<br/><a NAME="E2:28_1"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>;<br/><a NAME="E3:28_1"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:28_1"/>
( <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">f</font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">w1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">s1</font>,<font color="Maroon">s2</font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">w2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">s1</font>,<font color="Maroon">s2</font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> implies <font color="Maroon">g</font> <a href="msualg_6.html#R1" target="_self">is_e.translation_of</a> <font color="Maroon">A</font>,<font color="Maroon">w1</font>,<font color="Maroon">w2</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:28"/>
<font color="Maroon">f</font> is    <a href="msualg_6.html#M2" target="_self">Translation</a> of <font color="Maroon">A</font>,<font color="Maroon">s1</font>,<font color="Maroon">s2</font>
 <b>by </b><i>, , , , </i>;<br/>


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