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<div class="add">

<b>let </b><font color="Maroon" title="c1">A</font> be  <a href="unialg_1.html#V6" title="UNIALG_1:attr.6">partial</a> <a href="unialg_1.html#V8" title="UNIALG_1:attr.8">non-empty</a>  <a href="unialg_1.html#L1" title="UNIALG_1:struct.1">UAStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">P</font> be    <a href="pua2mss1.html#M3" title="PUA2MSS1:mode.3">a_partition</a> of <font color="Maroon" title="c1">A</font>;<br/>



<b>assume </b><a NAME="E1:72"/><i><font color="Green" title="E42">A1</font></i>: 
<font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K9" title="PUA2MSS1:func.9">LimDomRel</a> <font color="Maroon" title="c1">A</font></span>)</span>
 ;<br/>

<b>set </b><font color="Maroon" title="c3">S1</font> =  <a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font>;<br/>
<b>let </b><font color="Maroon" title="c4">S2</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> non <a href="msualg_1.html#V2" title="MSUALG_1:attr.2">void</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ;<br/>

<b>given </b><font color="Maroon" title="c5">G</font> being    <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c4">S2</font>, <font color="Maroon" title="c6">Q</font> being    <a href="pua2mss1.html#M2" title="PUA2MSS1:mode.2">IndexedPartition</a> of the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c4">S2</font><b> such that </b><a NAME="E2:72"/><i><font color="Green" title="E43">A2</font></i>: 
<font color="Maroon" title="c1">A</font> <a href="pua2mss1.html#R6" title="PUA2MSS1:pred.6">can_be_characterized_by</a> <font color="Maroon" title="c4">S2</font>,<font color="Maroon" title="c5">G</font>,<font color="Maroon" title="c6">Q</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="pua2mss1.html#D17" target="_self" title="PUA2MSS1:def.17">PUA2MSS1:def 17</a><br/>

<a NAME="E3:72"/><i><font color="Green" title="E44">A3</font></i>: 
( the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c5">G</font> is    <a href="pua2mss1.html#M4" title="PUA2MSS1:mode.4">IndexedPartition</a> of <font color="Maroon" title="c1">A</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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 <font color="Maroon" title="c1">A</font> )
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E2:72"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="pua2mss1.html#D16" target="_self" title="PUA2MSS1:def.16">Def16</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">G</font> = <font color="Maroon" title="c5">G</font> as  <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="c4">S2</font> <b>by </b><i><a class="ref" href="msualg_1.html#D8" title="MSUALG_1:def.8">MSUALG_1:def 8</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">R</font> = the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c7">G</font> as    <a href="pua2mss1.html#M4" title="PUA2MSS1:mode.4">IndexedPartition</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="txt" href="pua2mss1.html#E2:72"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="pua2mss1.html#D16" target="_self" title="PUA2MSS1:def.16">Def16</a></i>;<br/>
<a NAME="E6:72"/><i><font color="Green" title="E45">A4</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c8">R</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="c4">S2</font>
 
<b>by </b><i><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="c9">SG</font> = the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c7">G</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">S2</font>, <a href="pua2mss1.html#K10" title="PUA2MSS1:func.10">rng</a> <font color="Maroon" title="c8">R</font> <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>
<a NAME="E8:72"/><i><font color="Green" title="E46">A5</font></i>: 
the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">o</font>,<font color="Olive" title="b2">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">o</font> is   <a href="pua2mss1.html#NM2" title="PUA2MSS1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">A</font>, <font color="Olive" title="b2">p</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c2">P</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  :  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Olive" title="b2">p</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Olive" title="b1">o</font>,<font color="Maroon" title="c1">A</font></span>)</span> </span>}</span> 
 
<b>by </b><i><a class="ref" href="pua2mss1.html#D15" target="_self" title="PUA2MSS1:def.15">Def15</a></i>;<br/>
<a NAME="E9:72"/><i><font color="Green" title="E47">A6</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font>
 
<b>by </b><i><a class="ref" href="pua2mss1.html#D15" target="_self" title="PUA2MSS1:def.15">Def15</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> $2;<br/>
<a NAME="E10:72"/><i><font color="Green" title="E48">A7</font></i>: 
 <a href="pua2mss1.html#K10" title="PUA2MSS1:func.10">rng</a> <font color="Maroon" title="c8">R</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c2">P</font>
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E1:72"><i><font color="Green" title="E42">A1</font></i></a>, <a class="ref" href="pua2mss1.html#T28" target="_self" title="PUA2MSS1:th.28">Th28</a></i>;<br/>
<a NAME="E11:72"/><b>then </b><i><font color="Green" title="E49">A8</font></i>: 
for <font color="Olive" title="b1">r</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pua2mss1.html#K10" title="PUA2MSS1:func.10">rng</a> <font color="Maroon" title="c8">R</font> holds <br/> ex <font color="Olive" title="b2">p</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">p</font>] )
 
<b>by </b><i><a class="ref" href="setfam_1.html#D2" title="SETFAM_1:def.2">SETFAM_1:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">em</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E13:72"/><i><font color="Green" title="E50">A9</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c10">em</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pua2mss1.html#K10" title="PUA2MSS1:func.10">rng</a> <font color="Maroon" title="c8">R</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">em</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">P</font> )
 <b>and </b><br/><a NAME="E14:72"/><i><font color="Green" title="E51">A10</font></i>: 
for <font color="Olive" title="b1">r</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pua2mss1.html#K10" title="PUA2MSS1:func.10">rng</a> <font color="Maroon" title="c8">R</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">r</font>,<font color="Maroon" title="c10">em</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">r</font>]
 <b>from </b><i><a class="ref" href="wellord2.html#S1" title="WELLORD2:sch.1">WELLORD2:sch 1</a>(<a class="txt" href="pua2mss1.html#E11:72"><i><font color="Green" title="E49">A8</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c11">em</font> = <font color="Maroon" title="c10">em</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="pua2mss1.html#K10" title="PUA2MSS1:func.10">rng</a> <font color="Maroon" title="c8">R</font>,<font color="Maroon" title="c2">P</font> <b>by </b><i><a class="txt" href="pua2mss1.html#E13:72"><i><font color="Green" title="E50">A9</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/>
<a NAME="E16:72"/>
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">em</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c8">R</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c8">R</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">em</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c8">R</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">em</font> )
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E13:72"><i><font color="Green" title="E50">A9</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a>, <a class="ref" href="relat_1.html#T47" title="RELAT_1:th.47">RELAT_1:47</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">f</font> = <font color="Maroon" title="c11">em</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c8">R</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">S2</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> <b>by </b><i><a class="txt" href="pua2mss1.html#E6:72"><i><font color="Green" title="E45">A4</font></i></a>, <a class="txt" href="pua2mss1.html#E9:72"><i><font color="Green" title="E47">A6</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c12">f</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="pua2mss1.html#D14" target="_self" title="PUA2MSS1:def.14">PUA2MSS1:def 14</a><br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">P</font> ex <font color="Olive" title="b2">o</font> being  <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c4">S2</font> st <br/>( $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">Q</font> <a href="pua2mss1.html#K5" title="PUA2MSS1:func.5">-index_of</a> $1</span>)</span>,<font color="Olive" title="b1">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">o</font> &amp;  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Olive" title="b2">o</font>,<font color="Maroon" title="c7">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Olive" title="b1">p</font> );<br/>
<a NAME="E18:72"/><i><font color="Green" title="E52">A11</font></i>: 
for <font color="Olive" title="b1">s2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">s2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c4">S2</font> holds <br/> ex <font color="Olive" title="b2">s1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">s2</font>,<font color="Olive" title="b2">s1</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">s2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:72_1"/>
<font color="Maroon" title="c13">s2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c4">S2</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c14">s2</font> = <font color="Maroon" title="c13">s2</font> as   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c4">S2</font> ;<br/>
<a NAME="E3:72_1"/>
<font color="Maroon" title="c9">SG</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</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="c14">s2</font></span>)</span> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="pua2mss1.html#K10" title="PUA2MSS1:func.10">rng</a> <font color="Maroon" title="c8">R</font>
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E2"><i><font color="Green" title="E2">Lm2</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c15">p</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">P</font><b> such that </b><br/><a NAME="E5:72_1"/><i><font color="Green" title="E53">A12</font></i>: 
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">SG</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</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="c14">s2</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c15">p</font>
 <b>by </b><i><a class="txt" href="pua2mss1.html#E10:72"><i><font color="Green" title="E48">A7</font></i></a>, <a class="ref" href="pua2mss1.html#T4" target="_self" title="PUA2MSS1:th.4">Th4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">p</font> = <font color="Maroon" title="c15">p</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c2">P</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>
<b>take </b><font color="Maroon" title="c17">s1</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">Q</font> <a href="pua2mss1.html#K5" title="PUA2MSS1:func.5">-index_of</a> <font color="Maroon" title="c14">s2</font></span>)</span>,<font color="Maroon" title="c16">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>

<b>reconsider </b><font color="Maroon" title="c18">o</font> = <font color="Maroon" title="c6">Q</font> <a href="pua2mss1.html#K5" title="PUA2MSS1:func.5">-index_of</a> <font color="Maroon" title="c14">s2</font> as   <a href="pua2mss1.html#NM2" title="PUA2MSS1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="txt" href="pua2mss1.html#E3:72"><i><font color="Green" title="E44">A3</font></i></a>, <a class="ref" href="pua2mss1.html#D4" target="_self" title="PUA2MSS1:def.4">Def4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c19">aa</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c14">s2</font>,<font color="Maroon" title="c7">G</font>;<br/>
<a NAME="E9:72_1"/><i><font color="Green" title="E54">A13</font></i>: 
<font color="Maroon" title="c19">aa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c14">s2</font>,<font color="Maroon" title="c7">G</font>
 
;<br/>
<a NAME="E10:72_1"/><i><font color="Green" title="E55">A14</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">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="c14">s2</font>,<font color="Maroon" title="c7">G</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="c14">s2</font>,<font color="Maroon" title="c7">G</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="E11:72_1"/><i><font color="Green" title="E56">A15</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c4">S2</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="c4">S2</font> &amp;  <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="c7">G</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="c4">S2</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E12:72_1"/>
the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <font color="Maroon" title="c7">G</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">o</font></span>)</span> is    <a href="pua2mss1.html#M2" title="PUA2MSS1:mode.2">IndexedPartition</a> of  <a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c18">o</font>,<font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E2:72"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="pua2mss1.html#D16" target="_self" title="PUA2MSS1:def.16">Def16</a></i>;<br/>
<a NAME="E13:72_1"/><b>then </b>
 <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#U5" title="MSUALG_1:sel.5">Charact</a> of <font color="Maroon" title="c7">G</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">o</font></span>)</span></span>)</span> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of  <a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c18">o</font>,<font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="ref" href="pua2mss1.html#D3" target="_self" title="PUA2MSS1:def.3">Def3</a></i>;<br/>
<a NAME="E14:72_1"/><b>then </b>
( the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <font color="Maroon" title="c7">G</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">o</font></span>)</span> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of  <a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c18">o</font>,<font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">s2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">o</font> )
 
<b>by </b><i><a class="ref" href="pua2mss1.html#D4" target="_self" title="PUA2MSS1:def.4">Def4</a>, <a class="ref" href="relat_1.html#T148" title="RELAT_1:th.148">RELAT_1:148</a></i>;<br/>
<a NAME="E15:72_1"/><b>then </b>
(  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c14">s2</font>,<font color="Maroon" title="c7">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <font color="Maroon" title="c7">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">s2</font> &amp; the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <font color="Maroon" title="c7">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">s2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <font color="Maroon" title="c7">G</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">o</font></span>)</span> &amp; the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <font color="Maroon" title="c7">G</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">o</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c18">o</font>,<font color="Maroon" title="c1">A</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E11:72_1"><i><font color="Green" title="E56">A15</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E16:72_1"/><b>then </b><i><font color="Green" title="E57">A16</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">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="c14">s2</font>,<font color="Maroon" title="c7">G</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c18">o</font>,<font color="Maroon" title="c1">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T25" title="RELAT_1:th.25">RELAT_1:25</a></i>;<br/>
<i><font color="Green" title="E58">A17</font></i>:  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c14">s2</font>,<font color="Maroon" title="c7">G</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="c7">G</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="c4">S2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">s2</font></span>)</span>
<b>by </b><i><a class="txt" href="pua2mss1.html#E11:72_1"><i><font color="Green" title="E56">A15</font></i></a>, <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="c9">SG</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</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="c14">s2</font></span>)</span></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="E18:72_1"/><b>then </b>
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c16">p</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c18">o</font>,<font color="Maroon" title="c1">A</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E5:72_1"><i><font color="Green" title="E53">A12</font></i></a>, <a class="txt" href="pua2mss1.html#E9:72_1"><i><font color="Green" title="E54">A13</font></i></a>, <a class="txt" href="pua2mss1.html#E10:72_1"><i><font color="Green" title="E55">A14</font></i></a>, <a class="txt" href="pua2mss1.html#E16:72_1"><i><font color="Green" title="E57">A16</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<b>hence </b><a NAME="E19:72_1"/>
<font color="Maroon" title="c17">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="pua2mss1.html#E8:72"><i><font color="Green" title="E46">A5</font></i></a></i>;<br/>

<b>take </b>
<font color="Maroon" title="c16">p</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c14">s2</font>
;<br/>

<b>thus </b><a NAME="E20:72_1"/>
( <font color="Maroon" title="c17">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">Q</font> <a href="pua2mss1.html#K5" title="PUA2MSS1:func.5">-index_of</a> <font color="Maroon" title="c13">s2</font></span>)</span>,<font color="Maroon" title="c16">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c13">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">s2</font> &amp;  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c14">s2</font>,<font color="Maroon" title="c7">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c16">p</font> )
 <b>by </b><i><a class="txt" href="pua2mss1.html#E5:72_1"><i><font color="Green" title="E53">A12</font></i></a>, <a class="txt" href="pua2mss1.html#E17:72_1"><i><font color="Green" title="E58">A17</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c13">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E20:72"/><i><font color="Green" title="E53">A18</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c13">g</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="c4">S2</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> &amp; ( for <font color="Olive" title="b1">s</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c4">S2</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">s</font>,<font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font>] ) )
 <b>from </b><i><a class="ref" href="wellord2.html#S1" title="WELLORD2:sch.1">WELLORD2:sch 1</a>(<a class="txt" href="pua2mss1.html#E18:72"><i><font color="Green" title="E52">A11</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c14">g</font> = <font color="Maroon" title="c13">g</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c4">S2</font>,the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> <b>by </b><i><a class="txt" href="pua2mss1.html#E20:72"><i><font color="Green" title="E53">A18</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c14">g</font>
;<br/>

<b>thus </b><a NAME="E22:72"/><i><font color="Green" title="E54">A19</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">f</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="c4">S2</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c14">g</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="c4">S2</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c12">f</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="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> )
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pua2mss1.html#D13" target="_self" title="PUA2MSS1:def.13">PUA2MSS1:def 13</a><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c15">c</font> be   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c4">S2</font>;<br/><b>set </b><font color="Maroon" title="c16">s</font> = the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c4">S2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">c</font>;<br/><a NAME="E1:72_2"/><i><font color="Green" title="E55">A20</font></i>: 
( <font color="Maroon" title="c8">R</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="c4">S2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c7">G</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="c4">S2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">c</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c4">S2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="c4">S2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">c</font></span>)</span> )
 <b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>;<br/><a NAME="E2:72_2"/>
<font color="Maroon" title="c8">R</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="c4">S2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">c</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pua2mss1.html#K10" title="PUA2MSS1:func.10">rng</a> <font color="Maroon" title="c8">R</font>
 <b>by </b><i><a class="txt" href="pua2mss1.html#E6:72"><i><font color="Green" title="E45">A4</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="E3:72_2"/><b>then </b>
<font color="Maroon" title="c8">R</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="c4">S2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">c</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">em</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c4">S2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">c</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="pua2mss1.html#E14:72"><i><font color="Green" title="E51">A10</font></i></a></i>;<br/><a NAME="E4:72_2"/><b>then </b><i><font color="Green" title="E56">A21</font></i>: 
<font color="Maroon" title="c8">R</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="c4">S2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">c</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="c4">S2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">c</font></span>)</span>
 <b>by </b><i><a class="txt" href="pua2mss1.html#E6:72"><i><font color="Green" title="E45">A4</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><b>consider </b><font color="Maroon" title="c17">p</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">P</font>, <font color="Maroon" title="c18">o</font> being   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c4">S2</font><b> such that </b><br/><a NAME="E6:72_2"/><i><font color="Green" title="E57">A22</font></i>: 
( <font color="Maroon" title="c14">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">Q</font> <a href="pua2mss1.html#K5" title="PUA2MSS1:func.5">-index_of</a> <font color="Maroon" title="c15">c</font></span>)</span>,<font color="Maroon" title="c17">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c15">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">o</font> &amp;  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c18">o</font>,<font color="Maroon" title="c7">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c17">p</font> )
 <b>by </b><i><a class="txt" href="pua2mss1.html#E20:72"><i><font color="Green" title="E53">A18</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c19">p</font> = <font color="Maroon" title="c17">p</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c2">P</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c20">s2</font> = <font color="Maroon" title="c6">Q</font> <a href="pua2mss1.html#K5" title="PUA2MSS1:func.5">-index_of</a> <font color="Maroon" title="c15">c</font> as   <a href="pua2mss1.html#NM2" title="PUA2MSS1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="txt" href="pua2mss1.html#E3:72"><i><font color="Green" title="E44">A3</font></i></a>, <a class="ref" href="pua2mss1.html#D4" target="_self" title="PUA2MSS1:def.4">Def4</a></i>;<br/><b>consider </b><font color="Maroon" title="c21">aa</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c18">o</font>,<font color="Maroon" title="c7">G</font>;<br/><a NAME="E10:72_2"/>
the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <font color="Maroon" title="c7">G</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">s2</font></span>)</span> is    <a href="pua2mss1.html#M2" title="PUA2MSS1:mode.2">IndexedPartition</a> of  <a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c20">s2</font>,<font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="txt" href="pua2mss1.html#E2:72"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="pua2mss1.html#D16" target="_self" title="PUA2MSS1:def.16">Def16</a></i>;<br/><a NAME="E11:72_2"/><b>then </b>
 <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#U5" title="MSUALG_1:sel.5">Charact</a> of <font color="Maroon" title="c7">G</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">s2</font></span>)</span></span>)</span> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of  <a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c20">s2</font>,<font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="ref" href="pua2mss1.html#D3" target="_self" title="PUA2MSS1:def.3">Def3</a></i>;<br/><a NAME="E12:72_2"/><b>then </b>
( <font color="Maroon" title="c18">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">s2</font> &amp;  <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="c7">G</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="c4">S2</font> &amp; the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <font color="Maroon" title="c7">G</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">s2</font></span>)</span> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of  <a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c20">s2</font>,<font color="Maroon" title="c1">A</font> )
 <b>by </b><i><a class="txt" href="pua2mss1.html#E6:72_2"><i><font color="Green" title="E57">A22</font></i></a>, <a class="ref" href="pua2mss1.html#D4" target="_self" title="PUA2MSS1:def.4">Def4</a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a>, <a class="ref" href="relat_1.html#T148" title="RELAT_1:th.148">RELAT_1:148</a></i>;<br/><a NAME="E13:72_2"/><b>then </b><i><font color="Green" title="E58">A23</font></i>: 
(  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c18">o</font>,<font color="Maroon" title="c7">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <font color="Maroon" title="c7">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">o</font> &amp; the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <font color="Maroon" title="c7">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <font color="Maroon" title="c7">G</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">s2</font></span>)</span> &amp; the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <font color="Maroon" title="c7">G</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">s2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c20">s2</font>,<font color="Maroon" title="c1">A</font></span>)</span> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><a NAME="E14:72_2"/><b>then </b><i><font color="Green" title="E59">A24</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">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="c18">o</font>,<font color="Maroon" title="c7">G</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="c18">o</font>,<font color="Maroon" title="c7">G</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">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="c18">o</font>,<font color="Maroon" title="c7">G</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c20">s2</font>,<font color="Maroon" title="c1">A</font></span>)</span> &amp; <font color="Maroon" title="c21">aa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c18">o</font>,<font color="Maroon" title="c7">G</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relat_1.html#T25" title="RELAT_1:th.25">RELAT_1:25</a></i>;<br/><a NAME="E15:72_2"/><b>then </b>
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c19">p</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c20">s2</font>,<font color="Maroon" title="c1">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="pua2mss1.html#E6:72_2"><i><font color="Green" title="E57">A22</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><a NAME="E16:72_2"/><b>then </b><i><font color="Green" title="E60">A25</font></i>: 
<span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c20">s2</font>,<font color="Maroon" title="c1">A</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c19">p</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">c</font></span>)</span>
 <b>by </b><i><a class="txt" href="pua2mss1.html#E6:72_2"><i><font color="Green" title="E57">A22</font></i></a>, <a class="ref" href="pua2mss1.html#D15" target="_self" title="PUA2MSS1:def.15">Def15</a></i>;<br/><a NAME="E17:72_2"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c15">c</font>,<font color="Maroon" title="c7">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c15">c</font>,<font color="Maroon" title="c7">G</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c15">c</font>,<font color="Maroon" title="c7">G</font></span>)</span>
 <b>by </b><i><a class="txt" href="pua2mss1.html#E6:72_2"><i><font color="Green" title="E57">A22</font></i></a>, <a class="txt" href="pua2mss1.html#E14:72_2"><i><font color="Green" title="E59">A24</font></i></a>, <a class="ref" href="relat_1.html#T146" title="RELAT_1:th.146">RELAT_1:146</a></i>;<br/><a NAME="E18:72_2"/><b>then </b>
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c15">c</font>,<font color="Maroon" title="c7">G</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c15">c</font>,<font color="Maroon" title="c7">G</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c19">p</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c15">c</font>,<font color="Maroon" title="c7">G</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c19">p</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c20">s2</font>,<font color="Maroon" title="c1">A</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c19">p</font></span>)</span> )
 <b>by </b><i><a class="txt" href="pua2mss1.html#E6:72_2"><i><font color="Green" title="E57">A22</font></i></a>, <a class="txt" href="pua2mss1.html#E13:72_2"><i><font color="Green" title="E58">A23</font></i></a>, <a class="ref" href="relat_1.html#T156" title="RELAT_1:th.156">RELAT_1:156</a>, <a class="ref" href="relat_1.html#T157" title="RELAT_1:th.157">RELAT_1:157</a></i>;<br/><a NAME="E19:72_2"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c15">c</font>,<font color="Maroon" title="c7">G</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c20">s2</font>,<font color="Maroon" title="c1">A</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c19">p</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/><a NAME="E20:72_2"/><b>then </b>
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c15">c</font>,<font color="Maroon" title="c7">G</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">c</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c15">c</font>,<font color="Maroon" title="c7">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">aa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c15">c</font>,<font color="Maroon" title="c7">G</font></span>)</span> )
 <b>by </b><i><a class="txt" href="pua2mss1.html#E6:72_2"><i><font color="Green" title="E57">A22</font></i></a>, <a class="txt" href="pua2mss1.html#E14:72_2"><i><font color="Green" title="E59">A24</font></i></a>, <a class="txt" href="pua2mss1.html#E16:72_2"><i><font color="Green" title="E60">A25</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E21:72_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c15">c</font>,<font color="Maroon" title="c7">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">aa</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="c7">G</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="c4">S2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">c</font> &amp; <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c15">c</font>,<font color="Maroon" title="c7">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">aa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">c</font></span>)</span> )
 ;<br/><a NAME="E22:72_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c4">S2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> &amp; <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <span class="p2">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> &amp; <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c15">c</font>,<font color="Maroon" title="c7">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">aa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c4">S2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">c</font> &amp; <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c15">c</font>,<font color="Maroon" title="c7">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">aa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <span class="p2">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">c</font> )
 <b>by </b><i><a class="txt" href="pua2mss1.html#E9:72"><i><font color="Green" title="E47">A6</font></i></a>, <a class="txt" href="pua2mss1.html#E1:72_2"><i><font color="Green" title="E55">A20</font></i></a>, <a class="txt" href="pua2mss1.html#E4:72_2"><i><font color="Green" title="E56">A21</font></i></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>;<br/><b>hence </b><a NAME="E23:72_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c4">S2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <span class="p2">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">c</font>
 <b>by </b><i><a class="ref" href="pua2mss1.html#T3" target="_self" title="PUA2MSS1:th.3">Th3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E24:72"/>
<font color="Maroon" title="c12">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c4">S2</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c14">g</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon" title="c15">o</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c16">p</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>;<br/><b>assume </b><a NAME="E1:72_3"/>
<font color="Maroon" title="c15">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c4">S2</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c17">s</font> = <font color="Maroon" title="c15">o</font> as   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c4">S2</font> ;<br/><b>assume </b><a NAME="E3:72_3"/><i><font color="Green" title="E55">A26</font></i>: 
<font color="Maroon" title="c16">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c4">S2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">o</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c18">q</font> = the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c4">S2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">s</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="c4">S2</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  ;<br/><b>reconsider </b><font color="Maroon" title="c19">r</font> = <font color="Maroon" title="c9">SG</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">q</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="pua2mss1.html#K10" title="PUA2MSS1:func.10">rng</a> <font color="Maroon" title="c8">R</font> <b>by </b><i><a class="txt" href="pua2mss1.html#E2"><i><font color="Green" title="E2">Lm2</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c20">p'</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">P</font>, <font color="Maroon" title="c21">o'</font> being   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c4">S2</font><b> such that </b><br/><a NAME="E7:72_3"/><i><font color="Green" title="E56">A27</font></i>: 
( <font color="Maroon" title="c14">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">Q</font> <a href="pua2mss1.html#K5" title="PUA2MSS1:func.5">-index_of</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c20">p'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c17">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">o'</font> &amp;  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c21">o'</font>,<font color="Maroon" title="c7">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c20">p'</font> )
 <b>by </b><i><a class="txt" href="pua2mss1.html#E20:72"><i><font color="Green" title="E53">A18</font></i></a></i>;<br/><a NAME="E8:72_3"/>
<font color="Maroon" title="c14">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon" title="c22">o1</font> being   <a href="pua2mss1.html#NM2" title="PUA2MSS1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c23">p1</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c2">P</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> <b> such that </b><br/><a NAME="E10:72_3"/><i><font color="Green" title="E57">A28</font></i>: 
( <font color="Maroon" title="c14">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c22">o1</font>,<font color="Maroon" title="c23">p1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c23">p1</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c22">o1</font>,<font color="Maroon" title="c1">A</font></span>)</span> )
 <b>by </b><i><a class="txt" href="pua2mss1.html#E8:72"><i><font color="Green" title="E46">A5</font></i></a></i>;<br/><a NAME="E11:72_3"/>
( <font color="Maroon" title="c20">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">p1</font> &amp; <font color="Maroon" title="c6">Q</font> <a href="pua2mss1.html#K5" title="PUA2MSS1:func.5">-index_of</a> <font color="Maroon" title="c17">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">o1</font> )
 <b>by </b><i><a class="txt" href="pua2mss1.html#E7:72_3"><i><font color="Green" title="E56">A27</font></i></a>, <a class="txt" href="pua2mss1.html#E10:72_3"><i><font color="Green" title="E57">A28</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E12:72_3"/><b>then </b><i><font color="Green" title="E58">A29</font></i>: 
the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">p'</font>
 <b>by </b><i><a class="txt" href="pua2mss1.html#E10:72_3"><i><font color="Green" title="E57">A28</font></i></a>, <a class="ref" href="pua2mss1.html#D15" target="_self" title="PUA2MSS1:def.15">Def15</a></i>;<br/> <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c21">o'</font>,<font color="Maroon" title="c7">G</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="c7">G</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="c18">q</font>
<b>by </b><i><a class="txt" href="pua2mss1.html#E7:72_3"><i><font color="Green" title="E56">A27</font></i></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c19">r</font>
<b>by </b><i><a class="ref" href="pboole.html#D19" title="PBOOLE:def.19">PBOOLE:def 19</a></i>
;<br/><a NAME="E14:72_3"/><b>then </b>
<font color="Maroon" title="c11">em</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c19">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">p'</font>
 <b>by </b><i><a class="txt" href="pua2mss1.html#E14:72"><i><font color="Green" title="E51">A10</font></i></a>, <a class="txt" href="pua2mss1.html#E7:72_3"><i><font color="Green" title="E56">A27</font></i></a>, <a class="ref" href="pua2mss1.html#T5" target="_self" title="PUA2MSS1:th.5">Th5</a></i>;<br/><b>hence </b><a NAME="E15:72_3"/>
<font color="Maroon" title="c12">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c16">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">o</font></span>)</span>
 <b>by </b><i><a class="txt" href="pua2mss1.html#E3:72_3"><i><font color="Green" title="E55">A26</font></i></a>, <a class="txt" href="pua2mss1.html#E12:72_3"><i><font color="Green" title="E58">A29</font></i></a>, <a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></i>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E26:72"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c12">f</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="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</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><br/>

<b>thus </b><a NAME="E27:72"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c12">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">s</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="E1:72_4"/>
<font color="Maroon" title="c15">s</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 <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c16">s</font> = <font color="Maroon" title="c15">s</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">P</font> <b>by </b><i><a class="ref" href="pua2mss1.html#D15" target="_self" title="PUA2MSS1:def.15">Def15</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c17">a</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c16">s</font>;<br/>
<a NAME="E4:72_4"/><i><font color="Green" title="E55">A30</font></i>: 
( <font color="Maroon" title="c8">R</font> <a href="pua2mss1.html#K5" title="PUA2MSS1:func.5">-index_of</a> <font color="Maroon" title="c17">a</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="c8">R</font> &amp; <font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">R</font> <a href="pua2mss1.html#K5" title="PUA2MSS1:func.5">-index_of</a> <font color="Maroon" title="c17">a</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="pua2mss1.html#D4" target="_self" title="PUA2MSS1:def.4">Def4</a></i>;<br/>
<a NAME="E5:72_4"/><b>then </b>
( <font color="Maroon" title="c8">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">R</font> <a href="pua2mss1.html#K5" title="PUA2MSS1:func.5">-index_of</a> <font color="Maroon" title="c17">a</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pua2mss1.html#K10" title="PUA2MSS1:func.10">rng</a> <font color="Maroon" title="c8">R</font> &amp; <font color="Maroon" title="c11">em</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">R</font> <a href="pua2mss1.html#K5" title="PUA2MSS1:func.5">-index_of</a> <font color="Maroon" title="c17">a</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">R</font> <a href="pua2mss1.html#K5" title="PUA2MSS1:func.5">-index_of</a> <font color="Maroon" title="c17">a</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:72_4"/><b>then </b>
( <font color="Maroon" title="c8">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">R</font> <a href="pua2mss1.html#K5" title="PUA2MSS1:func.5">-index_of</a> <font color="Maroon" title="c17">a</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">R</font> <a href="pua2mss1.html#K5" title="PUA2MSS1:func.5">-index_of</a> <font color="Maroon" title="c17">a</font></span>)</span> &amp; <font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">R</font> <a href="pua2mss1.html#K5" title="PUA2MSS1:func.5">-index_of</a> <font color="Maroon" title="c17">a</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c12">f</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c12">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">P</font> )
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E6:72"><i><font color="Green" title="E45">A4</font></i></a>, <a class="txt" href="pua2mss1.html#E14:72"><i><font color="Green" title="E51">A10</font></i></a>, <a class="txt" href="pua2mss1.html#E22:72"><i><font color="Green" title="E54">A19</font></i></a>, <a class="txt" href="pua2mss1.html#E4:72_4"><i><font color="Green" title="E55">A30</font></i></a>, <a class="ref" href="pua2mss1.html#D15" target="_self" title="PUA2MSS1:def.15">Def15</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E7:72_4"/>
<font color="Maroon" title="c15">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c12">f</font>
 <b>by </b><i><a class="txt" href="pua2mss1.html#E4:72_4"><i><font color="Green" title="E55">A30</font></i></a>, <a class="ref" href="pua2mss1.html#T3" target="_self" title="PUA2MSS1:th.3">Th3</a></i>;<br/>


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

<b>thus </b><a NAME="E28:72"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</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><br/>

<b>thus </b><a NAME="E29:72"/>
the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">g</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">s1</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="E1:72_5"/>
<font color="Maroon" title="c15">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c16">o</font> being   <a href="pua2mss1.html#NM2" title="PUA2MSS1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c17">p</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c2">P</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> <b> such that </b><br/><a NAME="E3:72_5"/><i><font color="Green" title="E55">A31</font></i>: 
( <font color="Maroon" title="c15">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">o</font>,<font color="Maroon" title="c17">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c17">p</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c16">o</font>,<font color="Maroon" title="c1">A</font></span>)</span> )
 <b>by </b><i><a class="txt" href="pua2mss1.html#E8:72"><i><font color="Green" title="E46">A5</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c18">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:72_5"/><i><font color="Green" title="E56">A32</font></i>: 
( <font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c17">p</font> &amp; <font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c16">o</font>,<font color="Maroon" title="c1">A</font></span>)</span> )
 <b>by </b><i><a class="txt" href="pua2mss1.html#E3:72_5"><i><font color="Green" title="E55">A31</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c19">h</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E7:72_5"/><i><font color="Green" title="E57">A33</font></i>: 
( <font color="Maroon" title="c18">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">h</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c19">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c17">p</font> &amp; ( 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="c17">p</font> holds <br/><font color="Maroon" title="c19">h</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> <font color="Maroon" title="c17">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) )
 <b>by </b><i><a class="txt" href="pua2mss1.html#E5:72_5"><i><font color="Green" title="E56">A32</font></i></a>, <a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c20">h</font> = <font color="Maroon" title="c19">h</font> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> <b>by </b><i><a class="txt" href="pua2mss1.html#E7:72_5"><i><font color="Green" title="E57">A33</font></i></a>, <a class="txt" href="pua2mss1.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/>
<a NAME="E9:72_5"/>
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c17">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c17">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c17">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_6.html#T10" title="FUNCT_6:th.10">FUNCT_6:10</a></i>;<br/>
<a NAME="E10:72_5"/><b>then </b>
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c17">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">P</font> &amp;  ex <font color="Olive" title="b1">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Maroon" title="c20">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c17">p</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c17">p</font> ) )
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E5:72_5"><i><font color="Green" title="E56">A32</font></i></a>, <a class="txt" href="pua2mss1.html#E7:72_5"><i><font color="Green" title="E57">A33</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E11:72_5"/><b>then </b>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c17">p</font> &amp;  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c17">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c17">p</font></span>)</span> &amp;  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c17">p</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">P</font> &amp;  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">P</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">A</font> )
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a>, <a class="ref" href="card_3.html#D4" title="CARD_3:def.4">CARD_3:def 4</a>, <a class="ref" href="zfmisc_1.html#T95" title="ZFMISC_1:th.95">ZFMISC_1:95</a></i>;<br/>
<a NAME="E12:72_5"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">h</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">A</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E13:72_5"/><b>then </b>
<font color="Maroon" title="c20">h</font> is    <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 <font color="Maroon" title="c1">A</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/>
<b>then consider </b><font color="Maroon" title="c21">r</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="pua2mss1.html#K10" title="PUA2MSS1:func.10">rng</a> <font color="Maroon" title="c8">R</font><b> such that </b><br/><a NAME="E15:72_5"/><i><font color="Green" title="E58">A34</font></i>: 
<font color="Maroon" title="c20">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c21">r</font>
 <b>by </b><i><a class="ref" href="pua2mss1.html#T7" target="_self" title="PUA2MSS1:th.7">Th7</a></i>;<br/>
<a NAME="E16:72_5"/><i><font color="Green" title="E59">A35</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">r</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c21">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pua2mss1.html#K10" title="PUA2MSS1:func.10">rng</a> <font color="Maroon" title="c8">R</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c17">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">P</font> )
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E15:72_5"><i><font color="Green" title="E58">A34</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<a NAME="E17:72_5"/><i><font color="Green" title="E60">A36</font></i>: 
 <a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c16">o</font>,<font color="Maroon" title="c1">A</font> <a href="pua2mss1.html#R2" title="PUA2MSS1:pred.2">is_exactly_partitable_wrt</a> <font color="Maroon" title="c2">P</font>
 
<b>by </b><i><a class="ref" href="pua2mss1.html#D11" target="_self" title="PUA2MSS1:def.11">Def11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_5_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c22">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:72_5_1"/>
<font color="Maroon" title="c22">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="c21">r</font>
 ;<br/><a NAME="E2:72_5_1"/><b>then </b><i><font color="Green" title="E61">A37</font></i>: 
( <font color="Maroon" title="c20">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x</font> &amp; <font color="Maroon" title="c20">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x</font> &amp; <font color="Maroon" title="c21">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c21">r</font> &amp; <font color="Maroon" title="c17">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c17">p</font> )
 <b>by </b><i><a class="txt" href="pua2mss1.html#E7:72_5"><i><font color="Green" title="E57">A33</font></i></a>, <a class="txt" href="pua2mss1.html#E15:72_5"><i><font color="Green" title="E58">A34</font></i></a>, <a class="txt" href="pua2mss1.html#E16:72_5"><i><font color="Green" title="E59">A35</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>then consider </b><font color="Maroon" title="c23">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:72_5_1"/><i><font color="Green" title="E62">A38</font></i>: 
( <font color="Maroon" title="c23">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c21">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c23">c</font> )
 <b>by </b><i><a class="txt" href="pua2mss1.html#E10:72"><i><font color="Green" title="E48">A7</font></i></a>, <a class="ref" href="setfam_1.html#D2" title="SETFAM_1:def.2">SETFAM_1:def 2</a></i>;<br/><b>thus </b><a NAME="E5:72_5_1"/>
<font color="Maroon" title="c21">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x</font>
 <b>by </b><i><a class="txt" href="pua2mss1.html#E2:72_5_1"><i><font color="Green" title="E61">A37</font></i></a>, <a class="txt" href="pua2mss1.html#E4:72_5_1"><i><font color="Green" title="E62">A38</font></i></a>, <a class="ref" href="pua2mss1.html#T3" target="_self" title="PUA2MSS1:th.3">Th3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E19:72_5"/><b>then </b><i><font color="Green" title="E61">A39</font></i>: 
(  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c21">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c17">p</font> &amp;  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c17">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c16">o</font>,<font color="Maroon" title="c1">A</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E3:72_5"><i><font color="Green" title="E55">A31</font></i></a>, <a class="txt" href="pua2mss1.html#E7:72_5"><i><font color="Green" title="E57">A33</font></i></a>, <a class="txt" href="pua2mss1.html#E16:72_5"><i><font color="Green" title="E59">A35</font></i></a>, <a class="txt" href="pua2mss1.html#E17:72_5"><i><font color="Green" title="E60">A36</font></i></a>, <a class="ref" href="pua2mss1.html#D10" target="_self" title="PUA2MSS1:def.10">Def10</a>, <a class="ref" href="card_3.html#T38" title="CARD_3:th.38">CARD_3:38</a></i>;<br/>
<a NAME="E20:72_5"/><b>then </b>
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c21">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c16">o</font>,<font color="Maroon" title="c1">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 consider </b><font color="Maroon" title="c22">s2</font> being   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c4">S2</font><b> such that </b><br/><a NAME="E22:72_5"/><i><font color="Green" title="E62">A40</font></i>: 
( the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c7">G</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</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="c22">s2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">r</font> &amp; <font color="Maroon" title="c22">s2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">o</font> )
 <b>by </b><i><a class="txt" href="pua2mss1.html#E2:72"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="pua2mss1.html#T33" target="_self" title="PUA2MSS1:th.33">Th33</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c23">p'</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">P</font>, <font color="Maroon" title="c24">o'</font> being   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c4">S2</font><b> such that </b><br/><a NAME="E24:72_5"/><i><font color="Green" title="E63">A41</font></i>: 
( <font color="Maroon" title="c14">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">Q</font> <a href="pua2mss1.html#K5" title="PUA2MSS1:func.5">-index_of</a> <font color="Maroon" title="c22">s2</font></span>)</span>,<font color="Maroon" title="c23">p'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c22">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">o'</font> &amp;  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c24">o'</font>,<font color="Maroon" title="c7">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c23">p'</font> )
 <b>by </b><i><a class="txt" href="pua2mss1.html#E20:72"><i><font color="Green" title="E53">A18</font></i></a></i>;<br/>
<a NAME="E25:72_5"/><i><font color="Green" title="E64">A42</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c4">S2</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="c4">S2</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 href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c22">s2</font>,<font color="Maroon" title="c7">G</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="c7">G</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="c4">S2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">s2</font></span>)</span>
<b>by </b><i><a class="txt" href="pua2mss1.html#E25:72_5"><i><font color="Green" title="E64">A42</font></i></a>, <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> <font color="Maroon" title="c21">r</font>
<b>by </b><i><a class="txt" href="pua2mss1.html#E22:72_5"><i><font color="Green" title="E62">A40</font></i></a>, <a class="ref" href="pboole.html#D19" title="PBOOLE:def.19">PBOOLE:def 19</a></i>
;<br/>
<a NAME="E27:72_5"/><b>then </b>
( <font color="Maroon" title="c23">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">em</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c21">r</font> &amp; <font color="Maroon" title="c17">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">em</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c21">r</font> &amp; <font color="Maroon" title="c6">Q</font> <a href="pua2mss1.html#K5" title="PUA2MSS1:func.5">-index_of</a> <font color="Maroon" title="c22">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">o</font> )
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E3:72"><i><font color="Green" title="E44">A3</font></i></a>, <a class="txt" href="pua2mss1.html#E14:72"><i><font color="Green" title="E51">A10</font></i></a>, <a class="txt" href="pua2mss1.html#E19:72_5"><i><font color="Green" title="E61">A39</font></i></a>, <a class="txt" href="pua2mss1.html#E22:72_5"><i><font color="Green" title="E62">A40</font></i></a>, <a class="txt" href="pua2mss1.html#E24:72_5"><i><font color="Green" title="E63">A41</font></i></a>, <a class="ref" href="pua2mss1.html#D4" target="_self" title="PUA2MSS1:def.4">Def4</a>, <a class="ref" href="pua2mss1.html#T5" target="_self" title="PUA2MSS1:th.5">Th5</a></i>;<br/>
<b>hence </b><a NAME="E28:72_5"/>
<font color="Maroon" title="c15">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">g</font>
 <b>by </b><i><a class="txt" href="pua2mss1.html#E20:72"><i><font color="Green" title="E53">A18</font></i></a>, <a class="txt" href="pua2mss1.html#E3:72_5"><i><font color="Green" title="E55">A31</font></i></a>, <a class="txt" href="pua2mss1.html#E24:72_5"><i><font color="Green" title="E63">A41</font></i></a>, <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>


</div>
