<?xml version="1.0"?>
<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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">P</font> being   <a href="pua2mss1.html#M3" title="PUA2MSS1:mode.3">a_partition</a> of <font color="Maroon" title="c1">A</font> holds  <font color="Maroon" title="c1">A</font> <a href="pua2mss1.html#R7" title="PUA2MSS1:pred.7">can_be_characterized_by</a>  <a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">P</font></span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">A</font> <a href="pua2mss1.html#R7" title="PUA2MSS1:pred.7">can_be_characterized_by</a>  <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><br/>



<b>set </b><font color="Maroon" title="c3">S</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/>
<a NAME="E1:70"/>
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K4" title="PUA2MSS1:func.4">id</a> <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> &amp; <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 <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="pua2mss1.html#D15" target="_self" title="PUA2MSS1:def.15">Def15</a>, <a class="ref" href="relat_1.html#T71" title="RELAT_1:th.71">RELAT_1:71</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">Z</font> =  <a href="pua2mss1.html#K4" title="PUA2MSS1:func.4">id</a> <font color="Maroon" title="c2">P</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="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="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</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>-&gt; </b>   <a href="relset_1.html#M2" title="RELSET_1:mode.2">Relation</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font> = <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <span class="p2">(<span class="default">$1 <a href="pua2mss1.html#K12" title="PUA2MSS1:func.12">`1</a> </span>)</span>,<font color="Maroon" title="c1">A</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<span class="default">$1 <a href="pua2mss1.html#K13" title="PUA2MSS1:func.13">`2</a> </span>)</span></span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c5">D</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of 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> such that </b><br/><a NAME="E4:70"/><i><font color="Green" title="E40">A1</font></i>: 
for <font color="Olive" title="b1">o</font> being  <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</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> holds  <font color="Maroon" title="c5">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">o</font>)
 <b>from </b><i><a class="ref" href="pboole.html#S5" title="PBOOLE:sch.5">PBOOLE:sch 5</a>();<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="pua2mss1.html#NM2" title="PUA2MSS1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">A</font>) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</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> : <font color="Olive" title="b1">a</font> <a href="pua2mss1.html#K12" title="PUA2MSS1:func.12">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 </span>}</span> ;<br/>
<b>consider </b><font color="Maroon" title="c6">Q</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of  <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> such that </b><br/><a NAME="E6:70"/><i><font color="Green" title="E41">A2</font></i>: 
for <font color="Olive" title="b1">o</font> being  <a href="pua2mss1.html#NM2" title="PUA2MSS1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">A</font> holds  <font color="Maroon" title="c6">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">o</font>)
 <b>from </b><i><a class="ref" href="pboole.html#S5" title="PBOOLE:sch.5">PBOOLE:sch 5</a>();<br/></i>
<a NAME="E7:70"/><i><font color="Green" title="E42">A3</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E8:70"/><i><font color="Green" title="E43">A4</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> &amp; 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="E9:70"/>
<font color="Maroon" title="c6">Q</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D15" title="FUNCT_1:def.15">FUNCT_1:def 15</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">Q</font> or  not <font color="Maroon" title="c6">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )</span><br/>

<b>assume </b><a NAME="E1:70_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">Q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Maroon" title="c6">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a></span><br/>

<b>then reconsider </b><font color="Maroon" title="c8">o</font> = <font color="Maroon" title="c7">x</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="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">y</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <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="c8">o</font>,<font color="Maroon" title="c1">A</font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">y</font> = <font color="Maroon" title="c9">y</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  ;<br/>
<a NAME="E5:70_1"/>
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c10">y</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> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">P</font> )
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E7:70_1"/><i><font color="Green" title="E44">A5</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">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="c10">y</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c10">y</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="c11">f</font> )
 <b>by </b><i><a class="ref" href="pua2mss1.html#T6" target="_self" title="PUA2MSS1:th.6">Th6</a></i>;<br/>
<a NAME="E8:70_1"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E9:70_1"/><b>then </b>
<font color="Maroon" title="c11">f</font> is   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E7:70_1"><i><font color="Green" title="E44">A5</font></i></a>, <a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></i>;<br/>
<a NAME="E10:70_1"/><b>then </b>
<font color="Maroon" title="c11">f</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">P</font>
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E7:70_1"><i><font color="Green" title="E44">A5</font></i></a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">f</font> = <font color="Maroon" title="c11">f</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/>
<a NAME="E12:70_1"/>
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c12">f</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="c8">o</font>,<font color="Maroon" title="c1">A</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E7:70_1"><i><font color="Green" title="E44">A5</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E13:70_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c12">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c12">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">o</font> &amp; <font color="Maroon" title="c6">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</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> : <font color="Olive" title="b1">a</font> <a href="pua2mss1.html#K12" title="PUA2MSS1:func.12">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">o</font> </span>}</span>  )
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E6:70"><i><font color="Green" title="E41">A2</font></i></a>, <a class="txt" href="pua2mss1.html#E8:70"><i><font color="Green" title="E43">A4</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<a NAME="E14:70_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c12">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="c7">x</font>
 
;<br/>
<b>hence </b><a NAME="E15:70_1"/>
not <font color="Maroon" title="c6">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c7">Q</font> = <font color="Maroon" title="c6">Q</font> as  <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/>
<a NAME="E11:70"/>
<font color="Maroon" title="c5">D</font> is    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">Z</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <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>,<font color="Maroon" title="c4">Z</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 <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c8">x</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> or <font color="Maroon" title="c5">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> is    <a href="relset_1.html#M2" title="RELSET_1:mode.2">Relation</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">Z</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <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></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">Z</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 <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></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> )</span><br/>

<b>assume </b><a NAME="E1:70_2"/><i><font color="Green" title="E44">A6</font></i>: 
<font color="Maroon" title="c8">x</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> is    <a href="relset_1.html#M2" title="RELSET_1:mode.2">Relation</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">Z</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <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></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">Z</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 <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></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span><br/>

<b>then consider </b><font color="Maroon" title="c9">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="c10">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="E3:70_2"/><i><font color="Green" title="E45">A7</font></i>: 
( <font color="Maroon" title="c8">x</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="c9">o</font>,<font color="Maroon" title="c10">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="c10">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="c9">o</font>,<font color="Maroon" title="c1">A</font></span>)</span> )
 <b>by </b><i><a class="txt" href="pua2mss1.html#E8:70"><i><font color="Green" title="E43">A4</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">xx</font> = <font color="Maroon" title="c8">x</font> as   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</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#E1:70_2"><i><font color="Green" title="E44">A6</font></i></a></i>;<br/>
<a NAME="E5:70_2"/><i><font color="Green" title="E46">A8</font></i>: 
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</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="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#K4" title="RELSET_1:func.4">dom</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="hidden.html#R1" title="HIDDEN:pred.1">=</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>;<br/>
<a NAME="E6:70_2"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c10">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">P</font>
 
;<br/>
<a NAME="E7:70_2"/><b>then </b>
( 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> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">Z</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="c10">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">Z</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c10">p</font></span>)</span> &amp; <font color="Maroon" title="c4">Z</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c10">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p</font> )
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E8:70"><i><font color="Green" title="E43">A4</font></i></a>, <a class="txt" href="pua2mss1.html#E3:70_2"><i><font color="Green" title="E45">A7</font></i></a>, <a class="ref" href="pua2mss1.html#D15" target="_self" title="PUA2MSS1:def.15">Def15</a>, <a class="ref" href="pboole.html#D19" title="PBOOLE:def.19">PBOOLE:def 19</a>, <a class="ref" href="relat_1.html#T79" title="RELAT_1:th.79">RELAT_1:79</a></i>;<br/>
<a NAME="E8:70_2"/><b>then </b><i><font color="Green" title="E47">A9</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">Z</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <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></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c10">p</font>
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E1:70_2"><i><font color="Green" title="E44">A6</font></i></a>, <a class="txt" href="pua2mss1.html#E5:70_2"><i><font color="Green" title="E46">A8</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/>
<a NAME="E9:70_2"/><i><font color="Green" title="E48">A10</font></i>: 
( <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c9">o</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="c10">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_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> &amp; <font color="Maroon" title="c11">xx</font> <a href="pua2mss1.html#K13" title="PUA2MSS1:func.13">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p</font> &amp; <font color="Maroon" title="c11">xx</font> <a href="pua2mss1.html#K12" title="PUA2MSS1:func.12">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">o</font> )
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E3:70_2"><i><font color="Green" title="E45">A7</font></i></a>, <a class="ref" href="pua2mss1.html#D15" target="_self" title="PUA2MSS1:def.15">Def15</a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<a NAME="E10:70_2"/><b>then </b><i><font color="Green" title="E49">A11</font></i>: 
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c9">o</font>,<font color="Maroon" title="c1">A</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c10">p</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c9">o</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="c10">p</font></span>)</span> &amp; <font color="Maroon" title="c5">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">xx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c9">o</font>,<font color="Maroon" title="c1">A</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c10">p</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E4:70"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="relat_1.html#T148" title="RELAT_1:th.148">RELAT_1:148</a></i>;<br/>
<a NAME="E11:70_2"/>
 <a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c9">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/>
<a NAME="E12:70_2"/><b>then </b>
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c10">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="c9">o</font>,<font color="Maroon" title="c1">A</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E3:70_2"><i><font color="Green" title="E45">A7</font></i></a>, <a class="ref" href="pua2mss1.html#D10" target="_self" title="PUA2MSS1:def.10">Def10</a></i>;<br/>
<a NAME="E13:70_2"/><b>then </b>
( <font color="Maroon" title="c4">Z</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 <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> 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> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c9">o</font>,<font color="Maroon" title="c1">A</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c10">p</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c10">p</font> )
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E8:70"><i><font color="Green" title="E43">A4</font></i></a>, <a class="txt" href="pua2mss1.html#E5:70_2"><i><font color="Green" title="E46">A8</font></i></a>, <a class="ref" href="relat_1.html#T79" title="RELAT_1:th.79">RELAT_1:79</a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>
<b>hence </b><a NAME="E14:70_2"/>
<font color="Maroon" title="c5">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> is    <a href="relset_1.html#M2" title="RELSET_1:mode.2">Relation</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">Z</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <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></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">Z</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 <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></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font>
 <b>by </b><i><a class="txt" href="pua2mss1.html#E8:70_2"><i><font color="Green" title="E47">A9</font></i></a>, <a class="txt" href="pua2mss1.html#E9:70_2"><i><font color="Green" title="E48">A10</font></i></a>, <a class="txt" href="pua2mss1.html#E10:70_2"><i><font color="Green" title="E49">A11</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c8">D</font> = <font color="Maroon" title="c5">D</font> as    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">Z</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <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>,<font color="Maroon" title="c4">Z</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 <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/>
<a NAME="E13:70"/><i><font color="Green" title="E44">A12</font></i>: 
 <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c7">Q</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 <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_3"><b>proof </b></a><div class="add">

<div><b>hereby </b> <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>,<a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> 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="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c7">Q</font></span>
<div class="add"><b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c7">Q</font> implies <font color="Maroon" title="c9">x</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> )</span><br/><b>assume </b><a NAME="E1:70_3_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c7">Q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">x</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></span><br/><b>then consider </b><font color="Maroon" title="c10">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:70_3_1"/><i><font color="Green" title="E45">A13</font></i>: 
( <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">Q</font> &amp; <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">y</font> )
 <b>by </b><i><a class="ref" href="card_5.html#T10" title="CARD_5:th.10">CARD_5:10</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">y</font> = <font color="Maroon" title="c10">y</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#E3:70_3_1"><i><font color="Green" title="E45">A13</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E5:70_3_1"/>
<font color="Maroon" title="c7">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</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> : <font color="Olive" title="b1">a</font> <a href="pua2mss1.html#K12" title="PUA2MSS1:func.12">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="pua2mss1.html#E6:70"><i><font color="Green" title="E41">A2</font></i></a></i>;<br/><a NAME="E6:70_3_1"/><b>then </b>
 ex <font color="Olive" title="b1">a</font> being  <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</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> st <br/>( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b1">a</font> <a href="pua2mss1.html#K12" title="PUA2MSS1:func.12">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y</font> )
 <b>by </b><i><a class="txt" href="pua2mss1.html#E3:70_3_1"><i><font color="Green" title="E45">A13</font></i></a></i>;<br/><b>hence </b><a NAME="E7:70_3_1"/>
<font color="Maroon" title="c9">x</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c9">x</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> or <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c7">Q</font> )</span><br/>

<b>assume </b><a NAME="E2:70_3"/>
<font color="Maroon" title="c9">x</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c7">Q</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c10">b</font> = <font color="Maroon" title="c9">x</font> as   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</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/>
<a NAME="E4:70_3"/>
<font color="Maroon" title="c7">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">b</font> <a href="pua2mss1.html#K12" title="PUA2MSS1:func.12">`1</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</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> : <font color="Olive" title="b1">a</font> <a href="pua2mss1.html#K12" title="PUA2MSS1:func.12">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font> <a href="pua2mss1.html#K12" title="PUA2MSS1:func.12">`1</a>  </span>}</span> 
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E6:70"><i><font color="Green" title="E41">A2</font></i></a></i>;<br/>
<a NAME="E5:70_3"/><b>then </b>
<font color="Maroon" title="c10">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">b</font> <a href="pua2mss1.html#K12" title="PUA2MSS1:func.12">`1</a> </span>)</span>
 
;<br/>
<b>hence </b><a NAME="E6:70_3"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c7">Q</font>
 <b>by </b><i><a class="txt" href="pua2mss1.html#E7:70"><i><font color="Green" title="E42">A3</font></i></a>, <a class="ref" href="card_5.html#T10" title="CARD_5:th.10">CARD_5:10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">x</font>, <font color="Maroon" title="c10">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">Q</font> &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">Q</font> &amp; <font color="Maroon" title="c9">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">y</font> implies <font color="Maroon" title="c7">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c7">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">y</font> )</span><br/><b>assume </b><a NAME="E1:70_4"/><i><font color="Green" title="E45">A14</font></i>: 
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">Q</font> &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">Q</font> &amp; <font color="Maroon" title="c9">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">y</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c7">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">y</font></span><br/><b>then reconsider </b><font color="Maroon" title="c11">a</font> = <font color="Maroon" title="c9">x</font>, <font color="Maroon" title="c12">b</font> = <font color="Maroon" title="c10">y</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="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><b>thus </b><a NAME="E3:70_4"/>
<font color="Maroon" title="c7">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c7">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">y</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:70_4_1"/>
<font color="Maroon" title="c7">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c7">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>then consider </b><font color="Maroon" title="c13">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:70_4_1"/><i><font color="Green" title="E46">A15</font></i>: 
( <font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> &amp; <font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">y</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E4:70_4_1"/>
( <font color="Maroon" title="c7">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</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> : <font color="Olive" title="b1">c</font> <a href="pua2mss1.html#K12" title="PUA2MSS1:func.12">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">a</font> </span>}</span>  &amp; <font color="Maroon" title="c7">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</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> : <font color="Olive" title="b1">c</font> <a href="pua2mss1.html#K12" title="PUA2MSS1:func.12">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">b</font> </span>}</span>  )
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E6:70"><i><font color="Green" title="E41">A2</font></i></a></i>;<br/>
<a NAME="E5:70_4_1"/><b>then </b>
(  ex <font color="Olive" title="b1">c1</font> being  <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</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> st <br/>( <font color="Maroon" title="c13">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c1</font> &amp; <font color="Olive" title="b1">c1</font> <a href="pua2mss1.html#K12" title="PUA2MSS1:func.12">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">a</font> ) &amp;  ex <font color="Olive" title="b1">c2</font> being  <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</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> st <br/>( <font color="Maroon" title="c13">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c2</font> &amp; <font color="Olive" title="b1">c2</font> <a href="pua2mss1.html#K12" title="PUA2MSS1:func.12">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">b</font> ) )
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E3:70_4_1"><i><font color="Green" title="E46">A15</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:70_4_1"/>
contradiction
 <b>by </b><i><a class="txt" href="pua2mss1.html#E1:70_4"><i><font color="Green" title="E45">A14</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c9">Q</font> = <font color="Maroon" title="c7">Q</font> as    <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 <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#E13:70"><i><font color="Green" title="E44">A12</font></i></a>, <a class="ref" href="pua2mss1.html#T15" target="_self" title="PUA2MSS1:th.15">Th15</a></i>;<br/>
<b>take </b><font color="Maroon" title="c10">G</font> =  <a href="msualg_1.html#G3" title="MSUALG_1:aggr.3">MSAlgebra</a>(# <font color="Maroon" title="c4">Z</font>,<font color="Maroon" title="c8">D</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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">P</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 <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> st <font color="Maroon" title="c1">A</font> <a href="pua2mss1.html#R6" title="PUA2MSS1:pred.6">can_be_characterized_by</a>  <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>,<font color="Maroon" title="c10">G</font>,<font color="Olive" title="b1">P</font></span><br/>

<b>take </b>
<font color="Maroon" title="c9">Q</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">A</font> <a href="pua2mss1.html#R6" title="PUA2MSS1:pred.6">can_be_characterized_by</a>  <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>,<font color="Maroon" title="c10">G</font>,<font color="Maroon" title="c9">Q</font></span><br/>

<a NAME="E16:70"/>
 <a href="pua2mss1.html#K4" title="PUA2MSS1:func.4">id</a> <font color="Maroon" title="c2">P</font> is    <a href="pua2mss1.html#M4" title="PUA2MSS1:mode.4">IndexedPartition</a> of <font color="Maroon" title="c1">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_5"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:70_5"/>
 <a href="pua2mss1.html#K3" title="PUA2MSS1:func.3">rng</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K4" title="PUA2MSS1:func.4">id</a> <font color="Maroon" title="c2">P</font></span>)</span> is    <a href="pua2mss1.html#M3" title="PUA2MSS1:mode.3">a_partition</a> of <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T71" title="RELAT_1:th.71">RELAT_1:71</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pua2mss1.html#D12" target="_self" title="PUA2MSS1:def.12">PUA2MSS1:def 12</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E17:70"/>
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c10">G</font> is    <a href="pua2mss1.html#M4" title="PUA2MSS1:mode.4">IndexedPartition</a> of <font color="Maroon" title="c1">A</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="pua2mss1.html#D16" target="_self" title="PUA2MSS1:def.16">PUA2MSS1:def 16</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">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> &amp; ( for <font color="Olive" title="b1">o</font> being  <a href="pua2mss1.html#NM2" title="PUA2MSS1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">A</font> holds  the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <font color="Maroon" title="c10">G</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">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="Olive" title="b1">o</font>,<font color="Maroon" title="c1">A</font> ) )</span><br/>

<b>thus </b><a NAME="E18:70"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">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="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">o</font> being  <a href="pua2mss1.html#NM2" title="PUA2MSS1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">A</font> holds  the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <font color="Maroon" title="c10">G</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">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="Olive" title="b1">o</font>,<font color="Maroon" title="c1">A</font></span><br/>

<b>let </b><font color="Maroon" title="c11">o</font> be   <a href="pua2mss1.html#NM2" title="PUA2MSS1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <font color="Maroon" title="c10">G</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">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="c11">o</font>,<font color="Maroon" title="c1">A</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c12">PP</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Olive" title="b1">p</font></span>)</span> where <font color="Olive" title="b1">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>  : verum </span>}</span>  as    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <b>by </b><i><a class="ref" href="pua2mss1.html#T9" target="_self" title="PUA2MSS1:th.9">Th9</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">QQ</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p4">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span>)</span> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c12">PP</font> : <font color="Olive" title="b1">a</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="c11">o</font>,<font color="Maroon" title="c1">A</font></span>)</span> </span>}</span>  as    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of  <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="c11">o</font>,<font color="Maroon" title="c1">A</font></span>)</span> <b>by </b><i><a class="ref" href="pua2mss1.html#T11" target="_self" title="PUA2MSS1:th.11">Th11</a></i>;<br/>
<b>set </b><font color="Maroon" title="c14">F</font> = the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <font color="Maroon" title="c10">G</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">o</font></span>)</span>;<br/>
<a NAME="E21:70"/><i><font color="Green" title="E45">A16</font></i>: 
( <font color="Maroon" title="c9">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pua2mss1.html#K3" title="PUA2MSS1:func.3">rng</a> <font color="Maroon" title="c9">Q</font> &amp;  <a href="pua2mss1.html#K3" title="PUA2MSS1:func.3">rng</a> <font color="Maroon" title="c9">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</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#E7:70"><i><font color="Green" title="E42">A3</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="E22:70"/><b>then </b>
( <font color="Maroon" title="c9">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">o</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;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">D</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 <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="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E23:70"/><b>then </b><i><font color="Green" title="E46">A17</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</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="c10">G</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">o</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">o</font> &amp; <font color="Maroon" title="c9">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E21:70"><i><font color="Green" title="E45">A16</font></i></a>, <a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c15">F</font> = the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <font color="Maroon" title="c10">G</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">o</font></span>)</span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c16">Qo</font> = <font color="Maroon" title="c9">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">o</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="pua2mss1.html#E21:70"><i><font color="Green" title="E45">A16</font></i></a>, <a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">RR</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c1">A</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">a</font></span>)</span> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c13">QQ</font> : verum </span>}</span>  as    <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="c11">o</font>,<font color="Maroon" title="c1">A</font> <b>by </b><i><a class="ref" href="pua2mss1.html#T12" target="_self" title="PUA2MSS1:th.12">Th12</a></i>;<br/>
<a NAME="E27:70"/><i><font color="Green" title="E47">A18</font></i>: 
<font color="Maroon" title="c9">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</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> : <font color="Olive" title="b1">a</font> <a href="pua2mss1.html#K12" title="PUA2MSS1:func.12">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">o</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E6:70"><i><font color="Green" title="E41">A2</font></i></a></i>;<br/>
<a NAME="E28:70"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">RR</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">y</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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">F</font> or <font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">RR</font> )</span><br/>

<b>assume </b><a NAME="E1:70_6"/>
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">RR</font></span><br/>

<b>then consider </b><font color="Maroon" title="c19">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:70_6"/><i><font color="Green" title="E48">A19</font></i>: 
( <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c15">F</font> &amp; <font color="Maroon" title="c18">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:70_6"/>
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <font color="Maroon" title="c10">G</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">o</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E3:70_6"><i><font color="Green" title="E48">A19</font></i></a>, <a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>;<br/>
<a NAME="E5:70_6"/><b>then </b>
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">o</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c20">a</font> being   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</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> such that </b><br/><a NAME="E7:70_6"/><i><font color="Green" title="E49">A20</font></i>: 
( <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">a</font> &amp; <font color="Maroon" title="c20">a</font> <a href="pua2mss1.html#K12" title="PUA2MSS1:func.12">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">o</font> )
 <b>by </b><i><a class="txt" href="pua2mss1.html#E27:70"><i><font color="Green" title="E47">A18</font></i></a></i>;<br/>
<a NAME="E8:70_6"/>
<font color="Maroon" title="c20">a</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="c21">s</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="c22">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="E10:70_6"/><i><font color="Green" title="E50">A21</font></i>: 
( <font color="Maroon" title="c20">a</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="c21">s</font>,<font color="Maroon" title="c22">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="c22">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="c21">s</font>,<font color="Maroon" title="c1">A</font></span>)</span> )
 <b>by </b><i><a class="txt" href="pua2mss1.html#E8:70"><i><font color="Green" title="E43">A4</font></i></a></i>;<br/>
<a NAME="E11:70_6"/><i><font color="Green" title="E51">A22</font></i>: 
( <font color="Maroon" title="c21">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">o</font> &amp; <font color="Maroon" title="c20">a</font> <a href="pua2mss1.html#K13" title="PUA2MSS1:func.13">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">p</font> &amp;  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c22">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">PP</font> &amp;  <a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c11">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="txt" href="pua2mss1.html#E7:70_6"><i><font color="Green" title="E49">A20</font></i></a>, <a class="txt" href="pua2mss1.html#E10:70_6"><i><font color="Green" title="E50">A21</font></i></a>, <a class="ref" href="pua2mss1.html#D11" target="_self" title="PUA2MSS1:def.11">Def11</a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<a NAME="E12:70_6"/><b>then </b>
( <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c22">p</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">QQ</font> &amp;  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c22">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="c11">o</font>,<font color="Maroon" title="c1">A</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E10:70_6"><i><font color="Green" title="E50">A21</font></i></a>, <a class="ref" href="pua2mss1.html#D10" target="_self" title="PUA2MSS1:def.10">Def10</a></i>;<br/>
<a NAME="E13:70_6"/><b>then </b>
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c22">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">QQ</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E14:70_6"/><b>then </b>
( <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c1">A</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c22">p</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">RR</font> &amp; <font color="Maroon" title="c18">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">a</font> )
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E3:70_6"><i><font color="Green" title="E48">A19</font></i></a>, <a class="txt" href="pua2mss1.html#E7:70_6"><i><font color="Green" title="E49">A20</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>
<b>hence </b><a NAME="E15:70_6"/>
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">RR</font>
 <b>by </b><i><a class="txt" href="pua2mss1.html#E4:70"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="pua2mss1.html#E7:70_6"><i><font color="Green" title="E49">A20</font></i></a>, <a class="txt" href="pua2mss1.html#E11:70_6"><i><font color="Green" title="E51">A22</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c18">F</font> = <font color="Maroon" title="c15">F</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c16">Qo</font>,<font color="Maroon" title="c17">RR</font> <b>by </b><i><a class="txt" href="pua2mss1.html#E23:70"><i><font color="Green" title="E46">A17</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/>
<a NAME="E30:70"/><i><font color="Green" title="E48">A23</font></i>: 
<font color="Maroon" title="c17">RR</font> <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="c18">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">RR</font> or <font color="Maroon" title="c19">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="c18">F</font> )</span><br/>

<b>assume </b><a NAME="E1:70_7"/>
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">RR</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c19">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="c18">F</font></span><br/>

<b>then consider </b><font color="Maroon" title="c20">a</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c13">QQ</font><b> such that </b><br/><a NAME="E3:70_7"/><i><font color="Green" title="E49">A24</font></i>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c1">A</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c20">a</font>
 ;<br/>
<a NAME="E4:70_7"/>
<font color="Maroon" title="c20">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">QQ</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c21">b</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c12">PP</font><b> such that </b><br/><a NAME="E6:70_7"/><i><font color="Green" title="E50">A25</font></i>: 
( <font color="Maroon" title="c20">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">b</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c1">A</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c21">b</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="c11">o</font>,<font color="Maroon" title="c1">A</font></span>)</span> )
 ;<br/>
<a NAME="E7:70_7"/>
<font color="Maroon" title="c21">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">PP</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c22">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="E9:70_7"/><i><font color="Green" title="E51">A26</font></i>: 
<font color="Maroon" title="c21">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c22">p</font>
 ;<br/>
<a NAME="E10:70_7"/>
 <a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c11">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/>
<a NAME="E11:70_7"/><b>then </b>
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c22">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="c11">o</font>,<font color="Maroon" title="c1">A</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E6:70_7"><i><font color="Green" title="E50">A25</font></i></a>, <a class="txt" href="pua2mss1.html#E9:70_7"><i><font color="Green" title="E51">A26</font></i></a>, <a class="ref" href="pua2mss1.html#D10" target="_self" title="PUA2MSS1:def.10">Def10</a></i>;<br/>
<a NAME="E12:70_7"/><b>then </b><i><font color="Green" title="E52">A27</font></i>: 
<font color="Maroon" title="c21">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">a</font>
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E6:70_7"><i><font color="Green" title="E50">A25</font></i></a>, <a class="txt" href="pua2mss1.html#E9:70_7"><i><font color="Green" title="E51">A26</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E13:70_7"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c22">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c22">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">o</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c22">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">p</font> )
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E8:70"><i><font color="Green" title="E43">A4</font></i></a>, <a class="txt" href="pua2mss1.html#E6:70_7"><i><font color="Green" title="E50">A25</font></i></a>, <a class="txt" href="pua2mss1.html#E9:70_7"><i><font color="Green" title="E51">A26</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<a NAME="E14:70_7"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c22">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">D</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c22">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">o</font> &amp; <font color="Maroon" title="c8">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c22">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">x</font> )
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E4:70"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="pua2mss1.html#E27:70"><i><font color="Green" title="E47">A18</font></i></a>, <a class="txt" href="pua2mss1.html#E3:70_7"><i><font color="Green" title="E49">A24</font></i></a>, <a class="txt" href="pua2mss1.html#E9:70_7"><i><font color="Green" title="E51">A26</font></i></a>, <a class="txt" href="pua2mss1.html#E12:70_7"><i><font color="Green" title="E52">A27</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>hence </b><a NAME="E15:70_7"/>
<font color="Maroon" title="c19">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="c18">F</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T73" title="FUNCT_1:th.73">FUNCT_1:73</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E31:70"/>
<font color="Maroon" title="c18">F</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">x1</font>, <font color="Maroon" title="c20">x2</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="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c19">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c18">F</font> or  not <font color="Maroon" title="c20">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c18">F</font> or  not <font color="Maroon" title="c18">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x2</font> or <font color="Maroon" title="c19">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">x2</font> )</span><br/>

<b>assume </b><a NAME="E1:70_8"/><i><font color="Green" title="E49">A28</font></i>: 
( <font color="Maroon" title="c19">x1</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> <font color="Maroon" title="c18">F</font> &amp; <font color="Maroon" title="c20">x2</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> <font color="Maroon" title="c18">F</font> &amp; <font color="Maroon" title="c18">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c19">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">x2</font></span><br/>

<b>then consider </b><font color="Maroon" title="c21">a1</font> being   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</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> such that </b><br/><a NAME="E3:70_8"/><i><font color="Green" title="E50">A29</font></i>: 
( <font color="Maroon" title="c19">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">a1</font> &amp; <font color="Maroon" title="c21">a1</font> <a href="pua2mss1.html#K12" title="PUA2MSS1:func.12">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">o</font> )
 <b>by </b><i><a class="txt" href="pua2mss1.html#E23:70"><i><font color="Green" title="E46">A17</font></i></a>, <a class="txt" href="pua2mss1.html#E27:70"><i><font color="Green" title="E47">A18</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c22">a2</font> being   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</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> such that </b><br/><a NAME="E5:70_8"/><i><font color="Green" title="E51">A30</font></i>: 
( <font color="Maroon" title="c20">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">a2</font> &amp; <font color="Maroon" title="c22">a2</font> <a href="pua2mss1.html#K12" title="PUA2MSS1:func.12">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">o</font> )
 <b>by </b><i><a class="txt" href="pua2mss1.html#E23:70"><i><font color="Green" title="E46">A17</font></i></a>, <a class="txt" href="pua2mss1.html#E27:70"><i><font color="Green" title="E47">A18</font></i></a>, <a class="txt" href="pua2mss1.html#E1:70_8"><i><font color="Green" title="E49">A28</font></i></a></i>;<br/>
<a NAME="E6:70_8"/>
<font color="Maroon" title="c21">a1</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="c23">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="c24">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="E8:70_8"/><i><font color="Green" title="E52">A31</font></i>: 
( <font color="Maroon" title="c21">a1</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="c23">o1</font>,<font color="Maroon" title="c24">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="c24">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="c23">o1</font>,<font color="Maroon" title="c1">A</font></span>)</span> )
 <b>by </b><i><a class="txt" href="pua2mss1.html#E8:70"><i><font color="Green" title="E43">A4</font></i></a></i>;<br/>
<a NAME="E9:70_8"/>
<font color="Maroon" title="c22">a2</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="c25">o2</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="c26">p2</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="E11:70_8"/><i><font color="Green" title="E53">A32</font></i>: 
( <font color="Maroon" title="c22">a2</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="c25">o2</font>,<font color="Maroon" title="c26">p2</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="c26">p2</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="c25">o2</font>,<font color="Maroon" title="c1">A</font></span>)</span> )
 <b>by </b><i><a class="txt" href="pua2mss1.html#E8:70"><i><font color="Green" title="E43">A4</font></i></a></i>;<br/>
<a NAME="E12:70_8"/>
( <font color="Maroon" title="c18">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">a1</font> &amp; <font color="Maroon" title="c18">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">a2</font> &amp; <font color="Maroon" title="c21">a1</font> <a href="pua2mss1.html#K13" title="PUA2MSS1:func.13">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">p1</font> &amp; <font color="Maroon" title="c22">a2</font> <a href="pua2mss1.html#K13" title="PUA2MSS1:func.13">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">p2</font> )
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E1:70_8"><i><font color="Green" title="E49">A28</font></i></a>, <a class="txt" href="pua2mss1.html#E3:70_8"><i><font color="Green" title="E50">A29</font></i></a>, <a class="txt" href="pua2mss1.html#E5:70_8"><i><font color="Green" title="E51">A30</font></i></a>, <a class="txt" href="pua2mss1.html#E8:70_8"><i><font color="Green" title="E52">A31</font></i></a>, <a class="txt" href="pua2mss1.html#E11:70_8"><i><font color="Green" title="E53">A32</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<a NAME="E13:70_8"/><b>then </b><i><font color="Green" title="E54">A33</font></i>: 
( <font color="Maroon" title="c18">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c1">A</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c24">p1</font></span>)</span> &amp; <font color="Maroon" title="c18">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c1">A</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c26">p2</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E4:70"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="pua2mss1.html#E3:70_8"><i><font color="Green" title="E50">A29</font></i></a>, <a class="txt" href="pua2mss1.html#E5:70_8"><i><font color="Green" title="E51">A30</font></i></a></i>;<br/>
<a NAME="E14:70_8"/><i><font color="Green" title="E55">A34</font></i>: 
(  <a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c11">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> &amp; <font color="Maroon" title="c11">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">o1</font> &amp; <font color="Maroon" title="c11">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">o2</font> )
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E3:70_8"><i><font color="Green" title="E50">A29</font></i></a>, <a class="txt" href="pua2mss1.html#E5:70_8"><i><font color="Green" title="E51">A30</font></i></a>, <a class="txt" href="pua2mss1.html#E8:70_8"><i><font color="Green" title="E52">A31</font></i></a>, <a class="txt" href="pua2mss1.html#E11:70_8"><i><font color="Green" title="E53">A32</font></i></a>, <a class="ref" href="pua2mss1.html#D11" target="_self" title="PUA2MSS1:def.11">Def11</a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<a NAME="E15:70_8"/><b>then </b>
(  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c24">p1</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="c11">o</font>,<font color="Maroon" title="c1">A</font></span>)</span> &amp;  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c26">p2</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="c11">o</font>,<font color="Maroon" title="c1">A</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E8:70_8"><i><font color="Green" title="E52">A31</font></i></a>, <a class="txt" href="pua2mss1.html#E11:70_8"><i><font color="Green" title="E53">A32</font></i></a>, <a class="ref" href="pua2mss1.html#D10" target="_self" title="PUA2MSS1:def.10">Def10</a></i>;<br/>
<a NAME="E16:70_8"/><b>then </b>
(  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c24">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c1">A</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c24">p1</font></span>)</span></span>)</span> &amp;  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c26">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c1">A</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c26">p2</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>
<b>hence </b><a NAME="E17:70_8"/>
<font color="Maroon" title="c19">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">x2</font>
 <b>by </b><i><a class="txt" href="pua2mss1.html#E3:70_8"><i><font color="Green" title="E50">A29</font></i></a>, <a class="txt" href="pua2mss1.html#E5:70_8"><i><font color="Green" title="E51">A30</font></i></a>, <a class="txt" href="pua2mss1.html#E8:70_8"><i><font color="Green" title="E52">A31</font></i></a>, <a class="txt" href="pua2mss1.html#E11:70_8"><i><font color="Green" title="E53">A32</font></i></a>, <a class="txt" href="pua2mss1.html#E13:70_8"><i><font color="Green" title="E54">A33</font></i></a>, <a class="txt" href="pua2mss1.html#E14:70_8"><i><font color="Green" title="E55">A34</font></i></a>, <a class="ref" href="pua2mss1.html#T2" target="_self" title="PUA2MSS1:th.2">Th2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E32:70"/>
the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <font color="Maroon" title="c10">G</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">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="c11">o</font>,<font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="txt" href="pua2mss1.html#E30:70"><i><font color="Green" title="E48">A23</font></i></a>, <a class="ref" href="pua2mss1.html#T16" target="_self" title="PUA2MSS1:th.16">Th16</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
