<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">A</font> be  <a href="unialg_1.html#V6">partial</a> <a href="unialg_1.html#V8">non-empty</a>  <a href="unialg_1.html#L1">UAStr</a> ;<br/><b>let </b><font color="Maroon">P</font> be    <a href="pua2mss1.html#M3" target="_self">a_partition</a> of <font color="Maroon">A</font>;<br/>



<b>assume </b><a NAME="E1:73"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K9" target="_self">LimDomRel</a> <font color="Maroon">A</font></span>)</span>
 ;<br/>

<b>set </b><font color="Maroon">S1</font> =  <a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">A</font>,<font color="Maroon">P</font>;<br/>
<b>let </b><font color="Maroon">S2</font> be  non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/>

<b>given </b><font color="Maroon">G</font> being    <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S2</font>, <font color="Maroon">Q</font> being    <a href="pua2mss1.html#M2" target="_self">IndexedPartition</a> of the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S2</font><b> such that </b><a NAME="E2:73"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">A</font> <a href="pua2mss1.html#R6" target="_self">can_be_characterized_by</a> <font color="Maroon">S2</font>,<font color="Maroon">G</font>,<font color="Maroon">Q</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="pua2mss1.html#D17" target="_self">PUA2MSS1:def 17</a><br/>

<a NAME="E3:73"/><i><font color="Green">E37</font></i>: 
( the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">G</font> is    <a href="pua2mss1.html#M4" target="_self">IndexedPartition</a> of <font color="Maroon">A</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">A</font> )
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">G</font> = <font color="Maroon">G</font> as  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S2</font> <b>by </b><i><a class="ref" href="msualg_1.html#D8">MSUALG_1:def 8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">R</font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">G</font> as    <a href="pua2mss1.html#M4" target="_self">IndexedPartition</a> of <font color="Maroon">A</font> <b>by </b><i>, </i>;<br/>
<a NAME="E6:73"/><i><font color="Green">E45</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">SG</font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">G</font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font>, <a href="pua2mss1.html#K10" target="_self">rng</a> <font color="Maroon">R</font> <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<a NAME="E8:73"/><i><font color="Green">E46</font></i>: 
the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">A</font>,<font color="Maroon">P</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">o</font>,<font color="Olive">p</font></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">o</font> is   <a href="pua2mss1.html#NM2">OperSymbol</a> of <font color="Maroon">A</font>, <font color="Olive">p</font> is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">P</font> <a href="finseq_2.html#K3">*</a>  :  <a href="card_3.html#K4">product</a> <font color="Olive">p</font> <a href="subset_1.html#R2">meets</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Olive">o</font>,<font color="Maroon">A</font></span>)</span> </span>}</span> 
 
<b>by </b><i><a class="ref" href="pua2mss1.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E9:73"/><i><font color="Green">E58</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">A</font>,<font color="Maroon">P</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="pua2mss1.html#T1" target="_self">Th1</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a<sub>2</sub></font>;<br/>
<a NAME="E10:73"/><i><font color="Green">E61</font></i>: 
 <a href="pua2mss1.html#K10" target="_self">rng</a> <font color="Maroon">R</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">P</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E11:73"/><b>then </b><i><font color="Green">E72</font></i>: 
for <font color="Olive">r</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="pua2mss1.html#K10" target="_self">rng</a> <font color="Maroon">R</font> holds <br/> ex <font color="Olive">p</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">r</font>,<font color="Olive">p</font>] )
 
<b>by </b><i><a class="ref" href="setfam_1.html#D2">SETFAM_1:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">em</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E13:73"/><i><font color="Green">E73</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">em</font> <a href="hidden.html#R1">=</a>  <a href="pua2mss1.html#K10" target="_self">rng</a> <font color="Maroon">R</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">em</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> )
 <b>and </b><br/><a NAME="E14:73"/><i><font color="Green">E74</font></i>: 
for <font color="Olive">r</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="pua2mss1.html#K10" target="_self">rng</a> <font color="Maroon">R</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">r</font>,<font color="Maroon">em</font> <a href="funct_1.html#K1">.</a> <font color="Olive">r</font>]
 <b>from </b><i><a class="ref" href="wellord2.html#S1">WELLORD2:sch 1</a>();<br/></i>
<b>reconsider </b><font color="Maroon">em</font> = <font color="Maroon">em</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="pua2mss1.html#K10" target="_self">rng</a> <font color="Maroon">R</font>,<font color="Maroon">P</font> <b>by </b><i>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<a NAME="E16:73"/>
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">em</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">R</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">R</font> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">em</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">R</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">em</font> )
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a>, <a class="ref" href="relat_1.html#T47">RELAT_1:47</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">em</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">R</font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font>,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">A</font>,<font color="Maroon">P</font></span>)</span> <b>by </b><i><a class="txt" href="pua2mss1.html#E6:73"><i><font color="Green">E45</font></i></a>, <a class="ref" href="pua2mss1.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<b>take </b>
<font color="Maroon">f</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="pua2mss1.html#D14" target="_self">PUA2MSS1:def 14</a><br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">P</font> ex <font color="Olive">o</font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S2</font> st <br/>( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="pua2mss1.html#K5" target="_self">-index_of</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,<font color="Olive">p</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">o</font> &amp;  <a href="msualg_1.html#K3">Args</a> <font color="Olive">o</font>,<font color="Maroon">G</font> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K4">product</a> <font color="Olive">p</font> );<br/>
<a NAME="E18:73"/><i><font color="Green">E75</font></i>: 
for <font color="Olive">s2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">s2</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S2</font> holds <br/> ex <font color="Olive">s1</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">s1</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">A</font>,<font color="Maroon">P</font></span>)</span> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">s2</font>,<font color="Olive">s1</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s2</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:73_1"/>
<font color="Maroon">s2</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S2</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">s2</font> = <font color="Maroon">s2</font> as   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S2</font> ;<br/>
<a NAME="E3:73_1"/>
<font color="Maroon">SG</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">s2</font></span>)</span> is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="pua2mss1.html#K10" target="_self">rng</a> <font color="Maroon">R</font>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">p</font> being    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">P</font><b> such that </b><br/><a NAME="E5:73_1"/><i><font color="Green">E78</font></i>: 
 <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><font color="Maroon">SG</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">s2</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">p</font>
 <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">P</font> <a href="finseq_2.html#K3">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<b>take </b><font color="Maroon">s1</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="pua2mss1.html#K5" target="_self">-index_of</a> <font color="Maroon">s2</font></span>)</span>,<font color="Maroon">p</font></span><a href="tarski.html#K4">]</a></span>;<br/>

<b>reconsider </b><font color="Maroon">o</font> = <font color="Maroon">Q</font> <a href="pua2mss1.html#K5" target="_self">-index_of</a> <font color="Maroon">s2</font> as   <a href="pua2mss1.html#NM2">OperSymbol</a> of <font color="Maroon">A</font> <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">aa</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">s2</font>,<font color="Maroon">G</font>;<br/>
<a NAME="E9:73_1"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">aa</font> <a href="hidden.html#R2">in</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">s2</font>,<font color="Maroon">G</font>
 
;<br/>
<a NAME="E10:73_1"/><i><font color="Green">E127</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">s2</font>,<font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">s2</font>,<font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E11:73_1"/><i><font color="Green">E130</font></i>: 
(  <a href="relset_1.html#K4">dom</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S2</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S2</font> &amp;  <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S2</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E12:73_1"/>
the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">G</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font></span>)</span> is    <a href="pua2mss1.html#M2" target="_self">IndexedPartition</a> of  <a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E13:73_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">G</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font></span>)</span></span>)</span> is    <a href="eqrel_1.html#M1">a_partition</a> of  <a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<a NAME="E14:73_1"/><b>then </b>
( the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">G</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font></span>)</span> is    <a href="eqrel_1.html#M1">a_partition</a> of  <a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font> &amp; <font color="Maroon">s2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> )
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T148">RELAT_1:148</a></i>;<br/>
<a NAME="E15:73_1"/><b>then </b>
(  <a href="msualg_1.html#K5">Den</a> <font color="Maroon">s2</font>,<font color="Maroon">G</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</font> &amp; the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">G</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font></span>)</span> &amp; the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">G</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="pua2mss1.html#T5" target="_self">Th5</a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E16:73_1"/><b>then </b><i><font color="Green">E131</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">s2</font>,<font color="Maroon">G</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T25">RELAT_1:25</a></i>;<br/>
<i><font color="Green">E132</font></i>:  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">s2</font>,<font color="Maroon">G</font> = 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">G</font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s2</font></span>)</span>
<b>by </b><i><a class="ref" href="pua2mss1.html#T5" target="_self">Th5</a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
 <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><font color="Maroon">SG</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">s2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D19">PBOOLE:def 19</a></i>
;<br/>
<a NAME="E18:73_1"/><b>then </b>
 <a href="card_3.html#K4">product</a> <font color="Maroon">p</font> <a href="subset_1.html#R2">meets</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="pua2mss1.html#T4" target="_self">Th4</a>, , , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>hence </b><a NAME="E19:73_1"/>
<font color="Maroon">s1</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">A</font>,<font color="Maroon">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="pua2mss1.html#E8:73"><i><font color="Green">E46</font></i></a></i>;<br/>

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

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

<b>thus </b><a NAME="E20:73_1"/>
( <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="pua2mss1.html#K5" target="_self">-index_of</a> <font color="Maroon">s2</font></span>)</span>,<font color="Maroon">p</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> &amp;  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">s2</font>,<font color="Maroon">G</font> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">p</font> )
 <b>by </b><i>, <a class="txt" href="pua2mss1.html#E9:73"><i><font color="Green">E58</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E20:73"/><i><font color="Green">E133</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S2</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">A</font>,<font color="Maroon">P</font></span>)</span> &amp; ( for <font color="Olive">s</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">s</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S2</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">s</font>,<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font>] ) )
 <b>from </b><i><a class="ref" href="wellord2.html#S1">WELLORD2:sch 1</a>(<a class="ref" href="pua2mss1.html#T3" target="_self">Th3</a>);<br/></i>
<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S2</font>,the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">A</font>,<font color="Maroon">P</font></span>)</span> <b>by </b><i>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<b>take </b>
<font color="Maroon">g</font>
;<br/>

<b>thus </b><a NAME="E22:73"/><i><font color="Green">E134</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S2</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">A</font>,<font color="Maroon">P</font></span>)</span> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">A</font>,<font color="Maroon">P</font></span>)</span> )
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pua2mss1.html#D13" target="_self">PUA2MSS1:def 13</a><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c</font> be   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S2</font>;<br/><b>set </b><font color="Maroon">s</font> = the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font>;<br/><a NAME="E1:73_2"/><i><font color="Green">E135</font></i>: 
( <font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">G</font> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span> )
 <b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>;<br/><a NAME="E2:73_2"/>
<font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="pua2mss1.html#K10" target="_self">rng</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="txt" href="pua2mss1.html#E6:73"><i><font color="Green">E45</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E3:73_2"/><b>then </b>
<font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">em</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default">the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E4:73_2"/><b>then </b><i><font color="Green">E136</font></i>: 
<font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span>
 <b>by </b><i><a class="txt" href="pua2mss1.html#E6:73"><i><font color="Green">E45</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><b>consider </b><font color="Maroon">p</font> being    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">P</font>, <font color="Maroon">o</font> being   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S2</font><b> such that </b><br/><a NAME="E6:73_2"/><i><font color="Green">E137</font></i>: 
( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="pua2mss1.html#K5" target="_self">-index_of</a> <font color="Maroon">c</font></span>)</span>,<font color="Maroon">p</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> &amp;  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">G</font> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">p</font> )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">P</font> <a href="finseq_2.html#K3">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/><b>reconsider </b><font color="Maroon">s2</font> = <font color="Maroon">Q</font> <a href="pua2mss1.html#K5" target="_self">-index_of</a> <font color="Maroon">c</font> as   <a href="pua2mss1.html#NM2">OperSymbol</a> of <font color="Maroon">A</font> <b>by </b><i>, </i>;<br/><b>consider </b><font color="Maroon">aa</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">G</font>;<br/><a NAME="E10:73_2"/>
the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">G</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</font></span>)</span> is    <a href="pua2mss1.html#M2" target="_self">IndexedPartition</a> of  <a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">s2</font>,<font color="Maroon">A</font>
 <b>by </b><i>, </i>;<br/><a NAME="E11:73_2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">G</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</font></span>)</span></span>)</span> is    <a href="eqrel_1.html#M1">a_partition</a> of  <a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">s2</font>,<font color="Maroon">A</font>
 <b>by </b><i/>;<br/><a NAME="E12:73_2"/><b>then </b>
( <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</font> &amp;  <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S2</font> &amp; the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">G</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</font></span>)</span> is    <a href="eqrel_1.html#M1">a_partition</a> of  <a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">s2</font>,<font color="Maroon">A</font> )
 <b>by </b><i>, , <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a>, <a class="ref" href="relat_1.html#T148">RELAT_1:148</a></i>;<br/><a NAME="E13:73_2"/><b>then </b><i><font color="Green">E138</font></i>: 
(  <a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">G</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> &amp; the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">G</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</font></span>)</span> &amp; the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">G</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">s2</font>,<font color="Maroon">A</font></span>)</span> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E14:73_2"/><b>then </b><i><font color="Green">E139</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">G</font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">G</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">s2</font>,<font color="Maroon">A</font></span>)</span> &amp; <font color="Maroon">aa</font> <a href="hidden.html#R2">in</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">G</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relat_1.html#T25">RELAT_1:25</a></i>;<br/><a NAME="E15:73_2"/><b>then </b>
 <a href="card_3.html#K4">product</a> <font color="Maroon">p</font> <a href="subset_1.html#R2">meets</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">s2</font>,<font color="Maroon">A</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E16:73_2"/><b>then </b><i><font color="Green">E140</font></i>: 
<span class="p1">(<span class="default"><a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">s2</font>,<font color="Maroon">A</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">p</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U3">ResultSort</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">A</font>,<font color="Maroon">P</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="pua2mss1.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E17:73_2"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c</font>,<font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c</font>,<font color="Maroon">G</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K3">Args</a> <font color="Maroon">c</font>,<font color="Maroon">G</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="relat_1.html#T146">RELAT_1:146</a></i>;<br/><a NAME="E18:73_2"/><b>then </b>
(  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c</font>,<font color="Maroon">G</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c</font>,<font color="Maroon">G</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">p</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c</font>,<font color="Maroon">G</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">p</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">s2</font>,<font color="Maroon">A</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">p</font></span>)</span> )
 <b>by </b><i>, <a class="txt" href="pua2mss1.html#E9:73"><i><font color="Green">E58</font></i></a>, <a class="ref" href="relat_1.html#T156">RELAT_1:156</a>, <a class="ref" href="relat_1.html#T157">RELAT_1:157</a></i>;<br/><a NAME="E19:73_2"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c</font>,<font color="Maroon">G</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">s2</font>,<font color="Maroon">A</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E20:73_2"/><b>then </b>
(  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c</font>,<font color="Maroon">G</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U3">ResultSort</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">A</font>,<font color="Maroon">P</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c</font>,<font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">aa</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c</font>,<font color="Maroon">G</font></span>)</span> )
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E21:73_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c</font>,<font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">aa</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">G</font> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font> &amp; <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c</font>,<font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">aa</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U3">ResultSort</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">A</font>,<font color="Maroon">P</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span> )
 ;<br/><a NAME="E22:73_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <span class="p1">(<span class="default">the <a href="msualg_1.html#U3">ResultSort</a> of <span class="p2">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">A</font>,<font color="Maroon">P</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">g</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c</font>,<font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">aa</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font> &amp; <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c</font>,<font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">aa</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3">ResultSort</a> of <span class="p2">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">A</font>,<font color="Maroon">P</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">g</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font> )
 <b>by </b><i><a class="ref" href="pua2mss1.html#T2" target="_self">Th2</a>, , <a class="ref" href="pua2mss1.html#T5" target="_self">Th5</a>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>;<br/><b>hence </b><a NAME="E23:73_2"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3">ResultSort</a> of <span class="p2">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">A</font>,<font color="Maroon">P</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">g</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E24:73"/>
<font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S2</font> <a href="funct_2.html#R2">=</a> the <a href="msualg_1.html#U3">ResultSort</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">A</font>,<font color="Maroon">P</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">o</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">p</font> be   <a href="funct_1.html#NM1">Function</a>;<br/><b>assume </b><a NAME="E1:73_3"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S2</font>
 ;<br/><b>then reconsider </b><font color="Maroon">s</font> = <font color="Maroon">o</font> as   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S2</font> ;<br/><b>assume </b><a NAME="E3:73_3"/><i><font color="Green">E170</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 ;<br/><b>reconsider </b><font color="Maroon">q</font> = the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> as    <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font> <a href="finseq_2.html#K3">*</a>  ;<br/><b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">SG</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">q</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="pua2mss1.html#K10" target="_self">rng</a> <font color="Maroon">R</font> <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">p'</font> being    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">P</font>, <font color="Maroon">o'</font> being   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S2</font><b> such that </b><br/><a NAME="E7:73_3"/><i><font color="Green">E171</font></i>: 
( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="pua2mss1.html#K5" target="_self">-index_of</a> <font color="Maroon">s</font></span>)</span>,<font color="Maroon">p'</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o'</font> &amp;  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o'</font>,<font color="Maroon">G</font> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">p'</font> )
 <b>by </b><i/>;<br/><a NAME="E8:73_3"/>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">A</font>,<font color="Maroon">P</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">o1</font> being   <a href="pua2mss1.html#NM2">OperSymbol</a> of <font color="Maroon">A</font>, <font color="Maroon">p1</font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">P</font> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E10:73_3"/><i><font color="Green">E172</font></i>: 
( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">p1</font></span><a href="tarski.html#K4">]</a></span> &amp;  <a href="card_3.html#K4">product</a> <font color="Maroon">p1</font> <a href="subset_1.html#R2">meets</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">o1</font>,<font color="Maroon">A</font></span>)</span> )
 <b>by </b><i><a class="txt" href="pua2mss1.html#E8:73"><i><font color="Green">E46</font></i></a></i>;<br/><a NAME="E11:73_3"/>
( <font color="Maroon">p'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">Q</font> <a href="pua2mss1.html#K5" target="_self">-index_of</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o1</font> )
 <b>by </b><i><a class="ref" href="pua2mss1.html#T5" target="_self">Th5</a>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E12:73_3"/><b>then </b><i><font color="Green">E174</font></i>: 
the <a href="msualg_1.html#U2">Arity</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">A</font>,<font color="Maroon">P</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font>
 <b>by </b><i>, <a class="ref" href="pua2mss1.html#T1" target="_self">Th1</a></i>;<br/> <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o'</font>,<font color="Maroon">G</font> = 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">G</font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font>
<b>by </b><i><a class="ref" href="pua2mss1.html#T5" target="_self">Th5</a>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
 <a href="card_3.html#K4">product</a> <font color="Maroon">r</font>
<b>by </b><i><a class="ref" href="pboole.html#D19">PBOOLE:def 19</a></i>
;<br/><a NAME="E14:73_3"/><b>then </b>
<font color="Maroon">em</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font>
 <b>by </b><i>, <a class="ref" href="pua2mss1.html#T5" target="_self">Th5</a>, </i>;<br/><b>hence </b><a NAME="E15:73_3"/>
<font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U2">Arity</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">A</font>,<font color="Maroon">P</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="pua2mss1.html#E9:73"><i><font color="Green">E58</font></i></a>, <a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E26:73"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">A</font>,<font color="Maroon">P</font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>thus </b><a NAME="E27:73"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">A</font>,<font color="Maroon">P</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_4"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:73_4"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">A</font>,<font color="Maroon">P</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">s</font> = <font color="Maroon">s</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">P</font> <b>by </b><i><a class="ref" href="pua2mss1.html#T1" target="_self">Th1</a></i>;<br/>
<b>consider </b><font color="Maroon">a</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">s</font>;<br/>
<a NAME="E4:73_4"/><i><font color="Green">E176</font></i>: 
( <font color="Maroon">R</font> <a href="pua2mss1.html#K5" target="_self">-index_of</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">R</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="pua2mss1.html#K5" target="_self">-index_of</a> <font color="Maroon">a</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E5:73_4"/><b>then </b>
( <font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="pua2mss1.html#K5" target="_self">-index_of</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="pua2mss1.html#K10" target="_self">rng</a> <font color="Maroon">R</font> &amp; <font color="Maroon">em</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="pua2mss1.html#K5" target="_self">-index_of</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="pua2mss1.html#K5" target="_self">-index_of</a> <font color="Maroon">a</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T23">FUNCT_1:23</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:73_4"/><b>then </b>
( <font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="pua2mss1.html#K5" target="_self">-index_of</a> <font color="Maroon">a</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="pua2mss1.html#K5" target="_self">-index_of</a> <font color="Maroon">a</font></span>)</span> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="pua2mss1.html#K5" target="_self">-index_of</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> )
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E6:73"><i><font color="Green">E45</font></i></a>, , <a class="ref" href="pua2mss1.html#T4" target="_self">Th4</a>, , <a class="ref" href="pua2mss1.html#T1" target="_self">Th1</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E7:73_4"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 <b>by </b><i>, </i>;<br/>


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

<b>thus </b><a NAME="E28:73"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">A</font>,<font color="Maroon">P</font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>thus </b><a NAME="E29:73"/>
the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">A</font>,<font color="Maroon">P</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s1</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:73_5"/>
<font color="Maroon">s1</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">A</font>,<font color="Maroon">P</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">o</font> being   <a href="pua2mss1.html#NM2">OperSymbol</a> of <font color="Maroon">A</font>, <font color="Maroon">p</font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">P</font> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E3:73_5"/><i><font color="Green">E178</font></i>: 
( <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,<font color="Maroon">p</font></span><a href="tarski.html#K4">]</a></span> &amp;  <a href="card_3.html#K4">product</a> <font color="Maroon">p</font> <a href="subset_1.html#R2">meets</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font></span>)</span> )
 <b>by </b><i><a class="txt" href="pua2mss1.html#E8:73"><i><font color="Green">E46</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:73_5"/><i><font color="Green">E181</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">p</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>consider </b><font color="Maroon">h</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E7:73_5"/><i><font color="Green">E182</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> holds <br/><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> ) )
 <b>by </b><i><a class="ref" href="pua2mss1.html#T5" target="_self">Th5</a>, <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">h</font> = <font color="Maroon">h</font> as   <a href="finseq_1.html#NM1">FinSequence</a> <b>by </b><i>, </i>;<br/>
<a NAME="E9:73_5"/>
 <a href="card_3.html#K4">product</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="card_3.html#K3">Union</a> <font color="Maroon">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_6.html#T10">FUNCT_6:10</a></i>;<br/>
<a NAME="E10:73_5"/><b>then </b>
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> &amp;  ex <font color="Olive">f</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">f</font> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K3">Union</a> <font color="Maroon">p</font> ) )
 
<b>by </b><i><a class="ref" href="pua2mss1.html#T5" target="_self">Th5</a>, , <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E11:73_5"/><b>then </b>
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K3">Union</a> <font color="Maroon">p</font> &amp;  <a href="card_3.html#K3">Union</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">p</font></span>)</span> &amp;  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">p</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">P</font> &amp;  <a href="setfam_1.html#K5">union</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> )
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a>, <a class="ref" href="card_3.html#D4">CARD_3:def 4</a>, <a class="ref" href="zfmisc_1.html#T95">ZFMISC_1:95</a></i>;<br/>
<a NAME="E12:73_5"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E13:73_5"/><b>then </b>
<font color="Maroon">h</font> is    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">r</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="pua2mss1.html#K10" target="_self">rng</a> <font color="Maroon">R</font><b> such that </b><br/><a NAME="E15:73_5"/><i><font color="Green">E183</font></i>: 
<font color="Maroon">h</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">r</font>
 <b>by </b><i/>;<br/>
<a NAME="E16:73_5"/><i><font color="Green">E192</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">r</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">r</font> <a href="tarski.html#R1">c=</a>  <a href="pua2mss1.html#K10" target="_self">rng</a> <font color="Maroon">R</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> )
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E9:73"><i><font color="Green">E58</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<a NAME="E17:73_5"/><i><font color="Green">E193</font></i>: 
 <a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font> <a href="pua2mss1.html#R2" target="_self">is_exactly_partitable_wrt</a> <font color="Maroon">P</font>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_5_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:73_5_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">r</font>
 ;<br/><a NAME="E2:73_5_1"/><b>then </b><i><font color="Green">E194</font></i>: 
( <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">r</font> &amp; <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">p</font> )
 <b>by </b><i>, <a class="txt" href="pua2mss1.html#E9:73"><i><font color="Green">E58</font></i></a>, , <a class="ref" href="card_3.html#T18">CARD_3:18</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>then consider </b><font color="Maroon">c</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:73_5_1"/><i><font color="Green">E195</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c</font> )
 <b>by </b><i>, <a class="ref" href="setfam_1.html#D2">SETFAM_1:def 2</a></i>;<br/><b>thus </b><a NAME="E5:73_5_1"/>
<font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="pua2mss1.html#E10:73"><i><font color="Green">E61</font></i></a>, <a class="ref" href="pua2mss1.html#T6" target="_self">Th6</a>, </i>;<br/></div><b>end;</b></div>
<a NAME="E19:73_5"/><b>then </b><i><font color="Green">E196</font></i>: 
(  <a href="card_3.html#K4">product</a> <font color="Maroon">r</font> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">p</font> &amp;  <a href="card_3.html#K4">product</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font></span>)</span> )
 
<b>by </b><i>, , , , , <a class="ref" href="card_3.html#T38">CARD_3:38</a></i>;<br/>
<a NAME="E20:73_5"/><b>then </b>
 <a href="card_3.html#K4">product</a> <font color="Maroon">r</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then consider </b><font color="Maroon">s2</font> being   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S2</font><b> such that </b><br/><a NAME="E22:73_5"/><i><font color="Green">E197</font></i>: 
( the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">G</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">s2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">s2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> )
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">p'</font> being    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">P</font>, <font color="Maroon">o'</font> being   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S2</font><b> such that </b><br/><a NAME="E24:73_5"/><i><font color="Green">E198</font></i>: 
( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="pua2mss1.html#K5" target="_self">-index_of</a> <font color="Maroon">s2</font></span>)</span>,<font color="Maroon">p'</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o'</font> &amp;  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o'</font>,<font color="Maroon">G</font> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">p'</font> )
 <b>by </b><i/>;<br/>
<a NAME="E25:73_5"/><i><font color="Green">E199</font></i>: 
 <a href="relset_1.html#K4">dom</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S2</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S2</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
 <a href="msualg_1.html#K3">Args</a> <font color="Maroon">s2</font>,<font color="Maroon">G</font> = 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">G</font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s2</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
 <a href="card_3.html#K4">product</a> <font color="Maroon">r</font>
<b>by </b><i><a class="ref" href="pua2mss1.html#T6" target="_self">Th6</a>, <a class="ref" href="pboole.html#D19">PBOOLE:def 19</a></i>
;<br/>
<a NAME="E27:73_5"/><b>then </b>
( <font color="Maroon">p'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">em</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">r</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">em</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">r</font> &amp; <font color="Maroon">Q</font> <a href="pua2mss1.html#K5" target="_self">-index_of</a> <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> )
 
<b>by </b><i>, , <a class="txt" href="pua2mss1.html#E10:73"><i><font color="Green">E61</font></i></a>, <a class="ref" href="pua2mss1.html#T6" target="_self">Th6</a>, <a class="ref" href="pua2mss1.html#T7" target="_self">Th7</a>, , </i>;<br/>
<b>hence </b><a NAME="E28:73_5"/>
<font color="Maroon">s1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font>
 <b>by </b><i>, , <a class="ref" href="pua2mss1.html#T7" target="_self">Th7</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


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


</div>
