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<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> <a href="catalg_1.html#NM1">Signature</a>;<br/>

<b>given </b><font color="Maroon">A</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E1:27_1_1"/><i><font color="Green">E37</font></i>: 
(  <a href="catalg_1.html#K2" target="_self">CatSign</a> <font color="Maroon">A</font> is    <a href="instalg1.html#M1">Subsignature</a> of <font color="Maroon">S</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">2 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="catalg_1.html#D6" target="_self">CATALG_1:def 6</a><br/>

<b>consider </b><font color="Maroon">s</font> being   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>;<br/>
<b>consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">p</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:27_1_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> 2 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">p</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T103">ZFMISC_1:103</a></i>;<br/>
<a NAME="E5:27_1_1"/>
2 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">A</font> <a href="finseq_2.html#K3">*</a>  :  <a href="finseq_1.html#K3">len</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a> 2 </span>}</span> 
 
<b>by </b><i><a class="ref" href="finseq_2.html#D4">FINSEQ_2:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">q</font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">A</font> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E7:27_1_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> 2 )
 <b>by </b><i/>;<br/>
<a NAME="E8:27_1_1"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 2
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E9:27_1_1"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E10:27_1_1"/><b>then </b><i><font color="Green">E45</font></i>: 
( <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">q</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">A'</font> = <font color="Maroon">A</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> 1 as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A'</font> <b>by </b><i/>;<br/>
<a NAME="E13:27_1_1"/>
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="finseq_1.html#K12">*&gt;</a></span> is    <a href="finseq_2.html#M2">Element</a> of 1 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A'</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T118">FINSEQ_2:118</a></i>;<br/>
<a NAME="E14:27_1_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">1 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K2" target="_self">CatSign</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">1 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">2</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">3 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T128">ZFMISC_1:128</a></i>;<br/>
<a NAME="E15:27_1_1"/><b>then </b>
( <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">1 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">2</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">3 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">1 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">2</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">3 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i>, <a class="ref" href="instalg1.html#T11">INSTALG1:11</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E16:27_1_1"/>
the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_1.html#D5">MSUALG_1:def 5</a><br/>


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