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<b>let </b><font color="Maroon">A</font> be  <a href="unialg_1.html#V6">partial</a> <a href="unialg_1.html#V8">non-empty</a>  <a href="unialg_1.html#L1">UAStr</a> ;<br/><b>let </b><font color="Maroon">S2</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">G</font> be    <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S2</font>;<br/><b>let </b><font color="Maroon">Q</font> be    <a href="pua2mss1.html#M2" target="_self">IndexedPartition</a> of the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S2</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:72"/><i><font color="Green">E29</font></i>: 
( the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">G</font> is    <a href="pua2mss1.html#M4" target="_self">IndexedPartition</a> of <font color="Maroon">A</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">A</font> )
 <b>and </b><br/><a NAME="E2:72"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">o</font> being  <a href="pua2mss1.html#NM2">OperSymbol</a> of <font color="Maroon">A</font> holds  the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">G</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="funct_1.html#K1">.</a> <font color="Olive">o</font></span>)</span> is    <a href="pua2mss1.html#M2" target="_self">IndexedPartition</a> of  <a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Olive">o</font>,<font color="Maroon">A</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="pua2mss1.html#D16" target="_self">PUA2MSS1:def 16</a><br/>

<b>reconsider </b><font color="Maroon">R</font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">G</font> as    <a href="pua2mss1.html#M4" target="_self">IndexedPartition</a> of <font color="Maroon">A</font> <b>by </b><i/>;<br/>
<a NAME="E4:72"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">SG</font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">G</font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font>, <a href="pua2mss1.html#K10" target="_self">rng</a> <font color="Maroon">R</font> <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<b>let </b><font color="Maroon">o</font> be   <a href="pua2mss1.html#NM2">OperSymbol</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">r</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="relat_1.html#K2">rng</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">G</font>;<br/>



<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">r</font> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K10" target="_self">rng</a> <font color="Maroon">R</font></span>)</span> <a href="finseq_2.html#K3">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<b>assume </b><a NAME="E7:72"/><i><font color="Green">E37</font></i>: 
 <a href="card_3.html#K4">product</a> <font color="Maroon">r</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon">P</font> = the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">G</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font></span>)</span> as    <a href="pua2mss1.html#M2" target="_self">IndexedPartition</a> of  <a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">h</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <font color="Maroon">p</font>;<br/>
<a NAME="E10:72"/>
<font color="Maroon">h</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">r</font>
 
;<br/>
<a NAME="E11:72"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">h</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">h</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/>
<a NAME="E12:72"/><b>then </b><i><font color="Green">E45</font></i>: 
( <font color="Maroon">P</font> <a href="pua2mss1.html#K5" target="_self">-index_of</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">h</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">h</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">P</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">h</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">h</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="pua2mss1.html#K5" target="_self">-index_of</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">h</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">h</font></span>)</span></span><a href="tarski.html#K4">]</a></span></span>)</span> )
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">Qo</font> = <font color="Maroon">Q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="pua2mss1.html#K3" target="_self">rng</a> <font color="Maroon">Q</font> <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E14:72"/><i><font color="Green">E46</font></i>: 
 <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S2</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E15:72"/><b>then </b><i><font color="Green">E58</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Qo</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<b>reconsider </b><font color="Maroon">s</font> = <font color="Maroon">P</font> <a href="pua2mss1.html#K5" target="_self">-index_of</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">h</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">h</font></span>)</span></span><a href="tarski.html#K4">]</a></span> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Qo</font> <b>by </b><i><a class="txt" href="pua2mss1.html#E12:72"><i><font color="Green">E45</font></i></a>, <a class="txt" href="pua2mss1.html#E14:72"><i><font color="Green">E46</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">SG</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">s</font></span>)</span> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="pua2mss1.html#K10" target="_self">rng</a> <font color="Maroon">R</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">q</font> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K10" target="_self">rng</a> <font color="Maroon">R</font></span>)</span> <a href="finseq_2.html#K3">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Q</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Olive">t</font></span>)</span> where <font color="Olive">t</font> is    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K10" target="_self">rng</a> <font color="Maroon">R</font></span>)</span> <a href="finseq_2.html#K3">*</a>  : verum </span>}</span>  as    <a href="eqrel_1.html#M1">a_partition</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> <a href="finseq_2.html#K3">*</a>  <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">s</font>
;<br/>

<a NAME="E20:72"/><i><font color="Green">E61</font></i>: 
 <a href="relset_1.html#K4">dom</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S2</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S2</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<i><font color="Green">E72</font></i>:  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">s</font>,<font color="Maroon">G</font> = 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">G</font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
 <a href="card_3.html#K4">product</a> <font color="Maroon">q</font>
<b>by </b><i><a class="ref" href="pboole.html#D19">PBOOLE:def 19</a></i>
;<br/>
<a NAME="E22:72"/><i><font color="Green">E73</font></i>: 
(  <a href="card_3.html#K4">product</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp;  <a href="card_3.html#K4">product</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> )
 
;<br/>
<a NAME="E23:72"/>
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="pua2mss1.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<a NAME="E24:72"/><b>then </b>
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K5">Den</a> <font color="Maroon">s</font>,<font color="Maroon">G</font>
 
;<br/>
<a NAME="E25:72"/><b>then </b>
<font color="Maroon">h</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">s</font>,<font color="Maroon">G</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E12:72"><i><font color="Green">E45</font></i></a>, <a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a></i>;<br/>
<a NAME="E26:72"/><b>then </b>
 <a href="card_3.html#K4">product</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">q</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E27:72"/>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">G</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E28:72"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 ;<br/>


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