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<div class="add">

<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> <a href="catalg_1.html#V2" target="_self">Categorial</a> <a href="catalg_1.html#V3" target="_self">delta-concrete</a> <a href="catalg_1.html#NM1">Signature</a>;<br/>

<b>consider </b><font color="Maroon">A</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:44"/><i><font color="Green">E37</font></i>: 
(  <a href="catalg_1.html#K3" 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> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:44"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">s</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">s</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> holds <br/> ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ex <font color="Olive">p</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">p</font></span><a href="tarski.html#K4">]</a></span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> &amp; <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"><font color="Olive">i</font></span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p3">(<span class="default"><a href="catalg_1.html#K4" target="_self">underlay</a> <font color="Maroon">S</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> )
 <b>and </b><br/><a NAME="E5:44"/>
for <font color="Olive">o</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> holds <br/> ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ex <font color="Olive">p</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( <font color="Olive">o</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">p</font></span><a href="tarski.html#K4">]</a></span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> &amp; <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"><font color="Olive">i</font></span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p3">(<span class="default"><a href="catalg_1.html#K4" target="_self">underlay</a> <font color="Maroon">S</font></span>)</span></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> )
 <b>by </b><i/>;<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">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">p</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E8:44"/><i><font color="Green">E42</font></i>: 
( <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">i</font>,<font color="Maroon">p</font></span><a href="tarski.html#K4">]</a></span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> &amp; <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"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p3">(<span class="default"><a href="catalg_1.html#K4" target="_self">underlay</a> <font color="Maroon">S</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> )
 <b>by </b><i/>;<br/>
<a NAME="E9:44"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0 &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> )
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T128">ZFMISC_1:128</a></i>;<br/>
<a NAME="E10:44"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 2
 
<b>by </b><i/>;<br/>
<a NAME="E11:44"/><b>then </b><i><font color="Green">E46</font></i>: 
2 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K4" target="_self">underlay</a> <font color="Maroon">S</font></span>)</span> <a href="tarski.html#R1">c=</a> 2 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font>
 
<b>by </b><i>, , , <a class="ref" href="zfmisc_1.html#T117">ZFMISC_1:117</a></i>;<br/>
<a NAME="E12:44"/><i><font color="Green">E74</font></i>: 
 <a href="catalg_1.html#K4" target="_self">underlay</a> <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:44_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="catalg_1.html#K4" target="_self">underlay</a> <font color="Maroon">S</font>
 ;<br/>

<a NAME="E2:44_1"/><b>then </b>
<span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="hidden.html#R2">in</a> 2 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K4" target="_self">underlay</a> <font color="Maroon">S</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:44_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:44"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="catalg_1.html#K4" target="_self">underlay</a> <font color="Maroon">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:44_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:44_2"/><b>then </b>
<span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="hidden.html#R2">in</a> 2 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:44_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="finseq_1.html#K10">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="tarski.html#K2">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="finseq_2.html#T147">FINSEQ_2:147</a>, <a class="ref" href="zfmisc_1.html#T128">ZFMISC_1:128</a></i>;<br/>
<a NAME="E4:44_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="finseq_1.html#K10">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="xboole_0.html#K2">\/</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="finseq_1.html#K10">*&gt;</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E5:44_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="catalg_1.html#K4" target="_self">underlay</a> <font color="Maroon">S</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E14:44"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="catalg_1.html#K4" target="_self">underlay</a> <font color="Maroon">S</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E15:44"/>
<font color="Maroon">S</font> is    <a href="catalg_1.html#M1" target="_self">CatSignature</a> of  <a href="catalg_1.html#K4" target="_self">underlay</a> <font color="Maroon">S</font>
 <b>by </b><i>, </i>;<br/>


</div>
