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<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/>

<b>assume </b><a NAME="E1:74_1_1"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">g</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">g</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/>for <font color="Olive">p</font> being  <a href="finseq_1.html#NM1">FinSequence</a>  st <font color="Olive">p</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="Olive">g</font> holds <br/> ex <font color="Olive">f</font> being  <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">p</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="Olive">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="Olive">g</font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<font color="Olive">f</font></span><a href="tarski.html#K4">]</a></span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="circcomb.html#D9" target="_self">CIRCCOMB:def 9</a><br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">a<sub>1</sub></font> <a href="mcart_1.html#K2">`2</a> ;<br/>
<div><i><font color="Green">E33</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">g</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">p</font> be    <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="finseq_2.html#K3">*</a> ;<br/><b>assume </b><a NAME="E1:74_1_1_1"/>
( <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> &amp; <font color="Maroon">p</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> )
 ;<br/><b>then consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</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> <b> such that </b><br/><a NAME="E3:74_1_1_1"/><i><font color="Green">E38</font></i>: 
<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="Maroon">f</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="circcomb.html#E8:74_1_1"><i><font color="Green">E45</font></i></a></i>;<br/><b>thus </b><a NAME="E4:74_1_1_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">g</font>,<font color="Maroon">p</font>) is   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</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> 
 <b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">A</font> being  <a href="msualg_1.html#V4">strict</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E4:74_1_1"/><i><font color="Green">E42</font></i>: 
( the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="pboole.html#R6">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="margrel1.html#K6">BOOLEAN</a>  &amp; ( for <font color="Olive">g</font> being   <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">p</font> being   <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="finseq_2.html#K3">*</a>   st <font color="Olive">g</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> &amp; <font color="Olive">p</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="Olive">g</font> holds <br/>the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">g</font>,<font color="Olive">p</font>) ) )
 <b>from </b><i><a class="ref" href="circcomb.html#S2" target="_self">CIRCCOMB:sch 2</a>(<a class="ref" href="circcomb.html#T4" target="_self">Th4</a>);<br/></i>
<b>take </b>
<font color="Maroon">A</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="circcomb.html#D11" target="_self">CIRCCOMB:def 11</a><br/>

<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#D10" target="_self">CIRCCOMB:def 10</a><br/>

<b>assume </b><a NAME="E5:74_1_1"/><i><font color="Green">E43</font></i>: 
<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>
 ;<br/>

<b>then reconsider </b><font color="Maroon">p</font> = 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> as    <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="finseq_2.html#K3">*</a>  <b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</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> <b> such that </b><br/><a NAME="E8:74_1_1"/><i><font color="Green">E45</font></i>: 
<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="Maroon">f</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="circcomb.html#E8:74_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="circcomb.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E9:74_1_1"/>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="mcart_1.html#K2">`2</a> 
 
<b>by </b><i><a class="ref" href="circcomb.html#T6" target="_self">Th6</a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<b>hence </b><a NAME="E10:74_1_1"/>
<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>,<span class="p2">(<span class="default">the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">g</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="circcomb.html#T5" target="_self">Th5</a>, <a class="ref" href="circcomb.html#T6" target="_self">Th6</a></i>;<br/>


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