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<div class="add">

<b>let </b><font color="Maroon" title="c1">MS</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> non <a href="msualg_1.html#V2" title="MSUALG_1:attr.2">void</a> <a href="msualg_1.html#V6" title="MSUALG_1:attr.6">segmental</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon" title="c2">A</font> be  <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">MS</font>;<br/><b>let </b><font color="Maroon" title="c3">B</font> be  <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of <font color="Maroon" title="c2">A</font>;<br/>





<a NAME="E1:18"/><i><font color="Green" title="E16">A1</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">MS</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E2:18"/><b>then </b><i><font color="Green" title="E17">A2</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">B</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">C</font> = the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">B</font> as   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c2">A</font> <b>by </b><i><a class="ref" href="msualg_2.html#D10" title="MSUALG_2:def.10">MSUALG_2:def 10</a></i>;<br/>
<a NAME="E4:18"/><i><font color="Green" title="E18">A3</font></i>: 
( <font color="Maroon" title="c4">C</font> is <a href="msualg_2.html#V3" title="MSUALG_2:attr.3">opers_closed</a> &amp; the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <font color="Maroon" title="c3">B</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_2.html#K4" title="MSUALG_2:func.4">Opers</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">C</font> )
 
<b>by </b><i><a class="ref" href="msualg_2.html#D10" title="MSUALG_2:def.10">MSUALG_2:def 10</a></i>;<br/>
<a NAME="E5:18"/><i><font color="Green" title="E19">A4</font></i>: 
 <a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# <span class="p1">(<span class="default"><a href="msualg_1.html#K11" title="MSUALG_1:func.11">the_sort_of</a> <font color="Maroon" title="c2">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_1.html#K12" title="MSUALG_1:func.12">the_charact_of</a> <font color="Maroon" title="c2">A</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="msualg_1.html#D19" title="MSUALG_1:def.19">MSUALG_1:def 19</a></i>;<br/>
<a NAME="E6:18"/><i><font color="Green" title="E20">A5</font></i>: 
 <a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# <span class="p1">(<span class="default"><a href="msualg_1.html#K11" title="MSUALG_1:func.11">the_sort_of</a> <font color="Maroon" title="c3">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_1.html#K12" title="MSUALG_1:func.12">the_charact_of</a> <font color="Maroon" title="c3">B</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="msualg_1.html#D19" title="MSUALG_1:def.19">MSUALG_1:def 19</a></i>;<br/>
<b>let </b><font color="Maroon" title="c5">S</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <font color="Maroon" title="c2">A</font></span>)</span>;<br/>

<b>assume </b><a NAME="E7:18"/><i><font color="Green" title="E21">A6</font></i>: 
<font color="Maroon" title="c5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <font color="Maroon" title="c3">B</font></span>)</span>
 ;<br/>

<b>set </b><font color="Maroon" title="c6">1A</font> =  <a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <font color="Maroon" title="c2">A</font>;<br/>
<b>let </b><font color="Maroon" title="c7">o</font> be   <a href="unialg_2.html#NM5" title="UNIALG_2:NM.5">operation</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <font color="Maroon" title="c2">A</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="unialg_2.html#D5" title="UNIALG_2:def.5">UNIALG_2:def 5</a><br/>

<b>thus </b><a NAME="E8:18"/>
<font color="Maroon" title="c5">S</font> <a href="unialg_2.html#R2" title="UNIALG_2:pred.2">is_closed_on</a> <font color="Maroon" title="c7">o</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">s</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c5">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="unialg_2.html#D4" title="UNIALG_2:def.4">UNIALG_2:def 4</a><br/>

<b>assume </b><a NAME="E1:18_1"/><i><font color="Green" title="E22">A7</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c7">o</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c9">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:18_1"/><i><font color="Green" title="E23">A8</font></i>: 
( <font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <font color="Maroon" title="c2">A</font></span>)</span> &amp; <font color="Maroon" title="c7">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:18_1"/>
<font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="mssublat.html#E5:18"><i><font color="Green" title="E19">A4</font></i></a>, <a class="txt" href="mssublat.html#E3:18_1"><i><font color="Green" title="E23">A8</font></i></a>, <a class="ref" href="msualg_1.html#D18" title="MSUALG_1:def.18">MSUALG_1:def 18</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">n</font> = <font color="Maroon" title="c9">n</font> as   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">MS</font> <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E6:18_1"/><i><font color="Green" title="E24">A9</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c1">MS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c1">MS</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:18_1"/><i><font color="Green" title="E25">A10</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c1">MS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c1">MS</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E8:18_1"/>
<font color="Maroon" title="c4">C</font> <a href="msualg_2.html#R1" title="MSUALG_2:pred.1">is_closed_on</a> <font color="Maroon" title="c10">n</font>
 
<b>by </b><i><a class="txt" href="mssublat.html#E4:18"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="msualg_2.html#D7" title="MSUALG_2:def.7">MSUALG_2:def 7</a></i>;<br/>
<a NAME="E9:18_1"/><b>then </b><i><font color="Green" title="E26">A11</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">n</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c1">MS</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c1">MS</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font>
 
<b>by </b><i><a class="ref" href="msualg_2.html#D6" title="MSUALG_2:def.6">MSUALG_2:def 6</a></i>;<br/>
<a NAME="E10:18_1"/><i><font color="Green" title="E27">A12</font></i>: 
<font color="Maroon" title="c7">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">n</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c1">MS</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:18_1_1"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">n</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c1">MS</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">n</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c1">MS</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">s</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon" title="c8">s</font>
;<br/>

<b>thus </b><a NAME="E1:18_1_1_1"/><i><font color="Green" title="E28">A13</font></i>: 
<font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">n</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c1">MS</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:18_1_1_1_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">n</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c10">n</font>,<font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>then </b><i><font color="Green" title="E29">A14</font></i>:  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">n</font>,<font color="Maroon" title="c2">A</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c1">MS</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font>
<b>by </b><i><a class="ref" href="msualg_1.html#D9" title="MSUALG_1:def.9">MSUALG_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c1">MS</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">n</font></span>)</span>
<b>by </b><i><a class="txt" href="mssublat.html#E6:18_1"><i><font color="Green" title="E24">A9</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
;<br/>
<b>reconsider </b><font color="Maroon" title="c11">An</font> = the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c1">MS</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">n</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">MS</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  ;<br/>
<a NAME="E4:18_1_1_1_1"/><i><font color="Green" title="E30">A15</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">n</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c11">An</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mssublat.html#E2:18_1_1_1_1"><i><font color="Green" title="E29">A14</font></i></a>, <a class="ref" href="pboole.html#D19" title="PBOOLE:def.19">PBOOLE:def 19</a></i>;<br/>
<b>set </b><font color="Maroon" title="c12">f</font> = the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c11">An</font>;<br/>
<a NAME="E5:18_1_1_1_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="unialg_1.html#T1" title="UNIALG_1:th.1">UNIALG_1:1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c13">d</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:18_1_1_1_1"/><i><font color="Green" title="E31">A16</font></i>: 
<font color="Maroon" title="c13">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">o</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E8:18_1_1_1_1"/>
<font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <font color="Maroon" title="c2">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="partfun1.html#T119" title="PARTFUN1:th.119">PARTFUN1:119</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">o1</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E10:18_1_1_1_1"/><i><font color="Green" title="E32">A17</font></i>: 
( <font color="Maroon" title="c14">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">o</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c14">o1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">o1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <font color="Maroon" title="c2">A</font></span>)</span> )
 <b>by </b><i><a class="ref" href="partfun1.html#D5" title="PARTFUN1:def.5">PARTFUN1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">d</font> = <font color="Maroon" title="c13">d</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <font color="Maroon" title="c2">A</font></span>)</span> <b>by </b><i><a class="txt" href="mssublat.html#E7:18_1_1_1_1"><i><font color="Green" title="E31">A16</font></i></a>, <a class="txt" href="mssublat.html#E10:18_1_1_1_1"><i><font color="Green" title="E32">A17</font></i></a>, <a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E12:18_1_1_1_1"/>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">S</font> &amp; <font color="Maroon" title="c5">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <font color="Maroon" title="c2">A</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E13:18_1_1_1_1"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <font color="Maroon" title="c2">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c16">s1</font> = <font color="Maroon" title="c8">s</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <font color="Maroon" title="c2">A</font></span>)</span> <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E15:18_1_1_1_1"/><i><font color="Green" title="E33">A18</font></i>: 
( <font color="Maroon" title="c7">o</font> is <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">s1</font> &amp; <font color="Maroon" title="c15">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">o</font> )
 
<b>by </b><i><a class="txt" href="mssublat.html#E1:18_1"><i><font color="Green" title="E22">A7</font></i></a>, <a class="txt" href="mssublat.html#E7:18_1_1_1_1"><i><font color="Green" title="E31">A16</font></i></a>, <a class="ref" href="unialg_1.html#D10" title="UNIALG_1:def.10">UNIALG_1:def 10</a></i>;<br/>
<i><font color="Green" title="E34">A19</font></i>: <font color="Maroon" title="c7">o</font> = 
the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font>
<b>by </b><i><a class="txt" href="mssublat.html#E5:18"><i><font color="Green" title="E19">A4</font></i></a>, <a class="txt" href="mssublat.html#E3:18_1"><i><font color="Green" title="E23">A8</font></i></a>, <a class="ref" href="msualg_1.html#D18" title="MSUALG_1:def.18">MSUALG_1:def 18</a></i>
<br/>.= 
 <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">n</font>,<font color="Maroon" title="c2">A</font>
<b>by </b><i><a class="ref" href="msualg_1.html#D11" title="MSUALG_1:def.11">MSUALG_1:def 11</a></i>
;<br/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">n</font>,<font color="Maroon" title="c2">A</font></span>)</span> = 
 <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c10">n</font>,<font color="Maroon" title="c2">A</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <font color="Maroon" title="c2">A</font></span>)</span>
<b>by </b><i><a class="txt" href="mssublat.html#E5:18"><i><font color="Green" title="E19">A4</font></i></a>, <a class="ref" href="msualg_1.html#T11" title="MSUALG_1:th.11">MSUALG_1:11</a></i>
;<br/>
<a NAME="E18:18_1_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c10">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mssublat.html#E15:18_1_1_1_1"><i><font color="Green" title="E33">A18</font></i></a>, <a class="txt" href="mssublat.html#E16:18_1_1_1_1"><i><font color="Green" title="E34">A19</font></i></a>, <a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/>
<b>then </b><i><font color="Green" title="E35">A20</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">s</font> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c10">n</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">An</font>
<b>by </b><i><a class="ref" href="msualg_1.html#D6" title="MSUALG_1:def.6">MSUALG_1:def 6</a></i>
;<br/>
<a NAME="E20:18_1_1_1_1"/><i><font color="Green" title="E36">A21</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c11">An</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1_1_5"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:18_1_1_1_1_5"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c11">An</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1_1_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:18_1_1_1_1_5_1"/><i><font color="Green" title="E37">A22</font></i>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">s</font>
 ;<br/>

<a NAME="E2:18_1_1_1_1_5_1"/><i><font color="Green" title="E38">A23</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">An</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">MS</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E3:18_1_1_1_1_5_1"/>
<font color="Maroon" title="c11">An</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">An</font>
 
<b>by </b><i><a class="txt" href="mssublat.html#E19:18_1_1_1_1"><i><font color="Green" title="E35">A20</font></i></a>, <a class="txt" href="mssublat.html#E1:18_1_1_1_1_5_1"><i><font color="Green" title="E37">A22</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:18_1_1_1_1_5_1"/><b>then </b>
<font color="Maroon" title="c11">An</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">MS</font>
 
<b>by </b><i><a class="txt" href="mssublat.html#E2:18_1_1_1_1_5_1"><i><font color="Green" title="E38">A23</font></i></a></i>;<br/>
<a NAME="E5:18_1_1_1_1_5_1"/><b>then </b>
<font color="Maroon" title="c11">An</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>hence </b><a NAME="E6:18_1_1_1_1_5_1"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c11">An</font></span>)</span>
 <b>by </b><i><a class="txt" href="mssublat.html#E19:18_1_1_1_1"><i><font color="Green" title="E35">A20</font></i></a>, <a class="txt" href="mssublat.html#E1:18_1_1_1_1_5_1"><i><font color="Green" title="E37">A22</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/>


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

<b>thus </b><a NAME="E2:18_1_1_1_1_5"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c11">An</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">s</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1_1_5_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:18_1_1_1_1_5_2"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c11">An</font></span>)</span>
 ;<br/>

<b>hence </b><a NAME="E2:18_1_1_1_1_5_2"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">s</font>
 <b>by </b><i><a class="txt" href="mssublat.html#E19:18_1_1_1_1"><i><font color="Green" title="E35">A20</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E21:18_1_1_1_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c11">An</font></span>)</span> holds <br/><font color="Maroon" title="c8">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c11">An</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:18_1_1_1_1_6"/><i><font color="Green" title="E37">A24</font></i>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c11">An</font></span>)</span>
 ;<br/>

<a NAME="E2:18_1_1_1_1_6"/><b>then </b>
<span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c11">An</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c11">An</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c18">fx</font> = <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c11">An</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> as   <a href="pboole.html#NM3" title="PBOOLE:NM.3">Component</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font> <b>by </b><i><a class="ref" href="funct_1.html#T25" title="FUNCT_1:th.25">FUNCT_1:25</a></i>;<br/>
<a NAME="E4:18_1_1_1_1_6"/><i><font color="Green" title="E38">A25</font></i>: 
<font color="Maroon" title="c18">fx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K11" title="MSUALG_1:func.11">the_sort_of</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="ref" href="msualg_1.html#D17" title="MSUALG_1:def.17">MSUALG_1:def 17</a></i>;<br/>
<a NAME="E5:18_1_1_1_1_6"/>
( <font color="Maroon" title="c8">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">s</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">S</font> )
 
<b>by </b><i><a class="txt" href="mssublat.html#E20:18_1_1_1_1"><i><font color="Green" title="E36">A21</font></i></a>, <a class="txt" href="mssublat.html#E1:18_1_1_1_1_6"><i><font color="Green" title="E37">A24</font></i></a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:18_1_1_1_1_6"/><b>then </b>
<font color="Maroon" title="c8">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">S</font>
 
;<br/>
<b>hence </b><a NAME="E7:18_1_1_1_1_6"/>
<font color="Maroon" title="c8">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c11">An</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font>
 <b>by </b><i><a class="txt" href="mssublat.html#E5:18"><i><font color="Green" title="E19">A4</font></i></a>, <a class="txt" href="mssublat.html#E4:18_1_1_1_1_6"><i><font color="Green" title="E38">A25</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E22:18_1_1_1_1"/><b>then </b><i><font color="Green" title="E37">A26</font></i>: 
<font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">n</font>,<font color="Maroon" title="c2">A</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mssublat.html#E4:18_1_1_1_1"><i><font color="Green" title="E30">A15</font></i></a>, <a class="txt" href="mssublat.html#E20:18_1_1_1_1"><i><font color="Green" title="E36">A21</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<a NAME="E23:18_1_1_1_1"/>
<font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c1">MS</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1_1_7"><b>proof </b></a><div class="add">

<a NAME="E1:18_1_1_1_1_7"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c1">MS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c1">MS</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>then </b><i><font color="Green" title="E38">A27</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c1">MS</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">An</font>
<b>by </b><i><a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c11">An</font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D19" title="PBOOLE:def.19">PBOOLE:def 19</a></i>
;<br/>
<a NAME="E3:18_1_1_1_1_7"/><i><font color="Green" title="E39">A28</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c11">An</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1_1_7_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:18_1_1_1_1_7_2"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c11">An</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1_1_7_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:18_1_1_1_1_7_2_1"/><i><font color="Green" title="E40">A29</font></i>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">s</font>
 ;<br/>

<a NAME="E2:18_1_1_1_1_7_2_1"/><i><font color="Green" title="E41">A30</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">An</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">MS</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E3:18_1_1_1_1_7_2_1"/>
<font color="Maroon" title="c11">An</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">An</font>
 
<b>by </b><i><a class="txt" href="mssublat.html#E19:18_1_1_1_1"><i><font color="Green" title="E35">A20</font></i></a>, <a class="txt" href="mssublat.html#E1:18_1_1_1_1_7_2_1"><i><font color="Green" title="E40">A29</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:18_1_1_1_1_7_2_1"/><b>then </b>
<font color="Maroon" title="c11">An</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">MS</font>
 
<b>by </b><i><a class="txt" href="mssublat.html#E2:18_1_1_1_1_7_2_1"><i><font color="Green" title="E41">A30</font></i></a></i>;<br/>
<a NAME="E5:18_1_1_1_1_7_2_1"/><b>then </b>
<font color="Maroon" title="c11">An</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">C</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>hence </b><a NAME="E6:18_1_1_1_1_7_2_1"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c11">An</font></span>)</span>
 <b>by </b><i><a class="txt" href="mssublat.html#E19:18_1_1_1_1"><i><font color="Green" title="E35">A20</font></i></a>, <a class="txt" href="mssublat.html#E1:18_1_1_1_1_7_2_1"><i><font color="Green" title="E40">A29</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/>


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

<b>thus </b><a NAME="E2:18_1_1_1_1_7_2"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c11">An</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">s</font>
 <b>by </b><i><a class="txt" href="mssublat.html#E19:18_1_1_1_1"><i><font color="Green" title="E35">A20</font></i></a>, <a class="ref" href="relat_1.html#T44" title="RELAT_1:th.44">RELAT_1:44</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:18_1_1_1_1_7"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">s</font> holds <br/><font color="Maroon" title="c8">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c11">An</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1_1_7_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:18_1_1_1_1_7_3"/><i><font color="Green" title="E40">A31</font></i>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">s</font>
 ;<br/>

<a NAME="E2:18_1_1_1_1_7_3"/><b>then </b><i><font color="Green" title="E41">A32</font></i>: 
( <font color="Maroon" title="c8">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">s</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">S</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E3:18_1_1_1_1_7_3"/><i><font color="Green" title="E42">A33</font></i>: 
( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">s</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">An</font> )
 
<b>by </b><i><a class="txt" href="mssublat.html#E3:18_1_1_1_1_7"><i><font color="Green" title="E39">A28</font></i></a>, <a class="txt" href="mssublat.html#E1:18_1_1_1_1_7_3"><i><font color="Green" title="E40">A31</font></i></a>, <a class="ref" href="relat_1.html#T44" title="RELAT_1:th.44">RELAT_1:44</a></i>;<br/>
<a NAME="E4:18_1_1_1_1_7_3"/><i><font color="Green" title="E43">A34</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">An</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">MS</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E5:18_1_1_1_1_7_3"/>
<font color="Maroon" title="c11">An</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">An</font>
 
<b>by </b><i><a class="txt" href="mssublat.html#E3:18_1_1_1_1_7_3"><i><font color="Green" title="E42">A33</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:18_1_1_1_1_7_3"/><b>then </b>
<font color="Maroon" title="c11">An</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">MS</font>
 
<b>by </b><i><a class="txt" href="mssublat.html#E4:18_1_1_1_1_7_3"><i><font color="Green" title="E43">A34</font></i></a></i>;<br/>
<a NAME="E7:18_1_1_1_1_7_3"/><b>then </b>
( <font color="Maroon" title="c11">An</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">C</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c11">An</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">An</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="mssublat.html#E3:18_1_1_1_1_7"><i><font color="Green" title="E39">A28</font></i></a>, <a class="txt" href="mssublat.html#E1:18_1_1_1_1_7_3"><i><font color="Green" title="E40">A31</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c18">canx</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c11">An</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> as   <a href="pboole.html#NM3" title="PBOOLE:NM.3">Component</a> of <font color="Maroon" title="c4">C</font> <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c11">An</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> = 
<font color="Maroon" title="c18">canx</font>

<br/>.= 
<font color="Maroon" title="c5">S</font>
<b>by </b><i><a class="txt" href="mssublat.html#E6:18"><i><font color="Green" title="E20">A5</font></i></a>, <a class="txt" href="mssublat.html#E7:18"><i><font color="Green" title="E21">A6</font></i></a>, <a class="ref" href="msualg_1.html#D17" title="MSUALG_1:def.17">MSUALG_1:def 17</a></i>
;<br/>
<b>hence </b><a NAME="E10:18_1_1_1_1_7_3"/>
<font color="Maroon" title="c8">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c11">An</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font>
 <b>by </b><i><a class="txt" href="mssublat.html#E2:18_1_1_1_1_7_3"><i><font color="Green" title="E41">A32</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:18_1_1_1_1_7"/>
<font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c1">MS</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font>
 <b>by </b><i><a class="txt" href="mssublat.html#E2:18_1_1_1_1_7"><i><font color="Green" title="E38">A27</font></i></a>, <a class="txt" href="mssublat.html#E3:18_1_1_1_1_7"><i><font color="Green" title="E39">A28</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E24:18_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">n</font>,<font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c1">MS</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mssublat.html#E22:18_1_1_1_1"><i><font color="Green" title="E37">A26</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E25:18_1_1_1_1"/>
<font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">n</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c1">MS</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>;<br/>


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

<a NAME="E2:18_1_1_1"/><i><font color="Green" title="E29">A35</font></i>: 
the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font>
 
<b>by </b><i><a class="txt" href="mssublat.html#E5:18"><i><font color="Green" title="E19">A4</font></i></a>, <a class="ref" href="msualg_1.html#D18" title="MSUALG_1:def.18">MSUALG_1:def 18</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">n</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c1">MS</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">s</font> = 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">n</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">s</font>
<b>by </b><i><a class="txt" href="mssublat.html#E1:18_1_1_1"><i><font color="Green" title="E28">A13</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
<br/>.= 
<font color="Maroon" title="c7">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">s</font>
<b>by </b><i><a class="txt" href="mssublat.html#E3:18_1"><i><font color="Green" title="E23">A8</font></i></a>, <a class="txt" href="mssublat.html#E2:18_1_1_1"><i><font color="Green" title="E29">A35</font></i></a>, <a class="ref" href="msualg_1.html#D11" title="MSUALG_1:def.11">MSUALG_1:def 11</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:18_1_1"/>
<font color="Maroon" title="c7">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">n</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c1">MS</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c11">crn</font> = <font color="Maroon" title="c4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c1">MS</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">n</font></span>)</span> as   <a href="pboole.html#NM3" title="PBOOLE:NM.3">Component</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">B</font> <b>by </b><i><a class="txt" href="mssublat.html#E1:18"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="mssublat.html#E2:18"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">com</font> = <font color="Maroon" title="c5">S</font> as   <a href="pboole.html#NM3" title="PBOOLE:NM.3">Component</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">B</font> <b>by </b><i><a class="txt" href="mssublat.html#E6:18"><i><font color="Green" title="E20">A5</font></i></a>, <a class="txt" href="mssublat.html#E7:18"><i><font color="Green" title="E21">A6</font></i></a>, <a class="ref" href="msualg_1.html#D17" title="MSUALG_1:def.17">MSUALG_1:def 17</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c1">MS</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font> = 
<font color="Maroon" title="c11">crn</font>
<b>by </b><i><a class="txt" href="mssublat.html#E7:18_1"><i><font color="Green" title="E25">A10</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c12">com</font>
<b>by </b><i><a class="ref" href="msualg_1.html#T10" title="MSUALG_1:th.10">MSUALG_1:10</a></i>
;<br/>
<b>hence </b><a NAME="E14:18_1"/>
<font color="Maroon" title="c7">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">S</font>
 <b>by </b><i><a class="txt" href="mssublat.html#E9:18_1"><i><font color="Green" title="E26">A11</font></i></a>, <a class="txt" href="mssublat.html#E10:18_1"><i><font color="Green" title="E27">A12</font></i></a></i>;<br/>


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


</div>
