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<b>let </b><font color="Maroon">S</font> be    <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/>

<b>assume </b><a NAME="E1:3_1_1"/><i><font color="Green">E25</font></i>: 
the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> <a href="hidden.html#R1">=</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/>

<b>hence </b>the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> = 
 <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="relset_1.html#T26">RELSET_1:26</a></i>
<br/>.= 
 <a href="partfun1.html#K6">id</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font>
<b>by </b><i>, <a class="ref" href="relset_1.html#T27">RELSET_1:27</a></i>
;<br/> <i><font color="Red">:: according to </font></i><a class="ref" href="circcomb.html#D7">CIRCCOMB:def 7</a><br/>

<b>thus </b><a NAME="E3:3_1_1"/>
<font color="Maroon">S</font> is <a href="circcomb.html#V2">gate`1=arity</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">g</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="circcomb.html#D8">CIRCCOMB:def 8</a><br/>

<b>thus </b><a NAME="E1:3_1_1_2"/>
(  not <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> or <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">g</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="circcomb.html#D9">CIRCCOMB:def 9</a><br/>

<b>thus </b><a NAME="E4:3_1_1"/>
(  not <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> or for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">g</font> or  ex <font color="Olive">b<sub>2</sub></font> being  <a href="relset_1.html#M2">M6</a>(<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="margrel1.html#K6">BOOLEAN</a> , <a href="margrel1.html#K6">BOOLEAN</a> ) st <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> ) )
 <b>by </b><i/>;<br/>


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