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

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>



<span class="kw">set </span><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"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pua2mss1.html#D15" target="_self" title="PUA2MSS1:def.15">Def15</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><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#NM1" title="PBOOLE:NM.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> ;<br/>
<span class="kw">deffunc </span><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>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default">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> </span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> = <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/>
<span class="kw">consider </span><font color="Maroon" title="c5">D</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">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><span class="kw"> such that </span><br/><a NAME="E4:70"/><span class="lab"><font color="Green" title="E40">A1</font></span>: 
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>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="pboole.html#S5" title="PBOOLE:sch.5">PBOOLE:sch 5</a>();<br/></span>
<span class="kw">deffunc </span><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>) <span class="kw">-&gt; </span>   <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/>
<span class="kw">consider </span><font color="Maroon" title="c6">Q</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.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><span class="kw"> such that </span><br/><a NAME="E6:70"/><span class="lab"><font color="Green" title="E41">A2</font></span>: 
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>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="pboole.html#S5" title="PBOOLE:sch.5">PBOOLE:sch 5</a>();<br/></span>
<a NAME="E7:70"/><span class="lab"><font color="Green" title="E42">A3</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>;<br/>
<a NAME="E8:70"/><span class="lab"><font color="Green" title="E43">A4</font></span>: 
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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pua2mss1.html#D15" target="_self" title="PUA2MSS1:def.15">Def15</a></span>;<br/>
<a NAME="E9:70"/><span class="lab"><font color="Green" title="E44">A5</font></span>: 
the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">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> <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#K1" title="RELSET_1:func.1">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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pua2mss1.html#D15" target="_self" title="PUA2MSS1:def.15">Def15</a></span>;<br/>
<a NAME="E10:70"/>
<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"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K1" title="RELAT_1:func.1">proj1</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/>

<span class="kw">assume </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">then </span><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>;<br/>
<span class="kw">consider </span><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#K1" title="RELSET_1:func.1">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/>
<span class="kw">reconsider </span><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"/><span class="lab"><font color="Green" title="E45">A6</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">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>
 
;<br/>
<a NAME="E6:70_1"/>
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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E8:70_1"/><span class="lab"><font color="Green" title="E46">A7</font></span>: 
 <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>
 <span class="kw">and </span><br/><a NAME="E9:70_1"/><span class="lab"><font color="Green" title="E47">A8</font></span>: 
 <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>
 <span class="kw">and </span><br/><a NAME="E10:70_1"/><span class="lab"><font color="Green" title="E48">A9</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:70_1"><span class="lab"><font color="Green" title="E45">A6</font></span></a>, <a class="ref" href="pua2mss1.html#T6" target="_self" title="PUA2MSS1:th.6">Th6</a></span>;<br/>
<a NAME="E11:70_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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E12:70_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">f</font> is   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:70_1"><span class="lab"><font color="Green" title="E46">A7</font></span></a>, <a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></span>;<br/>
<a NAME="E13:70_1"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:70_1"><span class="lab"><font color="Green" title="E47">A8</font></span></a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></span>;<br/>
<a NAME="E15:70_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#K1" title="RELSET_1:func.1">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:70_1"><span class="lab"><font color="Green" title="E48">A9</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<a NAME="E16:70_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A10</font></span>: 
<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="struct_0.html#U4" title="STRUCT_0:sel.4">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:70"><span class="lab"><font color="Green" title="E44">A5</font></span></a></span>;<br/>
<a NAME="E17:70_1"/><span class="lab"><font color="Green" title="E50">A11</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>;<br/>
<a NAME="E18:70_1"/>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:70"><span class="lab"><font color="Green" title="E41">A2</font></span></a></span>;<br/>
<a NAME="E19:70_1"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:70_1"><span class="lab"><font color="Green" title="E49">A10</font></span></a>, <a class="txt" href="#E17:70_1"><span class="lab"><font color="Green" title="E50">A11</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E20:70_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><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="E12:70"/>
<font color="Maroon" title="c5">D</font> is    <a href="pboole.html#M2" title="PBOOLE:mode.2">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="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">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="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">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"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="struct_0.html#U4" title="STRUCT_0:sel.4">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> 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="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<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#NK3" title="FUNCT_1:NK.3">*</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <span class="p4">(<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>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">Z</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <span class="p4">(<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>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:70_2"/><span class="lab"><font color="Green" title="E45">A12</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<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#NK3" title="FUNCT_1:NK.3">*</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <span class="p4">(<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>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">Z</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <span class="p4">(<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>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E3:70_2"/><span class="lab"><font color="Green" title="E46">A13</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:70_2"/><span class="lab"><font color="Green" title="E47">A14</font></span>: 
 <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#K1" title="RELSET_1:func.1">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:70"><span class="lab"><font color="Green" title="E44">A5</font></span></a></span>;<br/>
<span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_2"><span class="lab"><font color="Green" title="E45">A12</font></span></a></span>;<br/>
<a NAME="E6:70_2"/><span class="lab"><font color="Green" title="E48">A15</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">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>
 
;<br/>
<a NAME="E7:70_2"/><span class="lab"><font color="Green" title="E49">A16</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">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="struct_0.html#U4" title="STRUCT_0:sel.4">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E8:70_2"/><span class="lab"><font color="Green" title="E50">A17</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">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="E9:70_2"/><span class="lab"><font color="Green" title="E51">A18</font></span>: 
the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:70_2"><span class="lab"><font color="Green" title="E46">A13</font></span></a>, <a class="txt" href="#E4:70_2"><span class="lab"><font color="Green" title="E47">A14</font></span></a>, <a class="ref" href="pua2mss1.html#D15" target="_self" title="PUA2MSS1:def.15">Def15</a></span>;<br/>
<a NAME="E10:70_2"/><span class="lab"><font color="Green" title="E52">A19</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:70"><span class="lab"><font color="Green" title="E43">A4</font></span></a>, <a class="ref" href="pboole.html#D19" title="PBOOLE:def.19">PBOOLE:def 19</a></span>;<br/>
<a NAME="E11:70_2"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:70_2"><span class="lab"><font color="Green" title="E50">A17</font></span></a>, <a class="ref" href="relat_1.html#T79" title="RELAT_1:th.79">RELAT_1:79</a></span>;<br/>
<a NAME="E12:70_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A20</font></span>: 
<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="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_2"><span class="lab"><font color="Green" title="E45">A12</font></span></a>, <a class="txt" href="#E7:70_2"><span class="lab"><font color="Green" title="E49">A16</font></span></a>, <a class="txt" href="#E9:70_2"><span class="lab"><font color="Green" title="E51">A18</font></span></a>, <a class="txt" href="#E10:70_2"><span class="lab"><font color="Green" title="E52">A19</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>;<br/>
<a NAME="E13:70_2"/><span class="lab"><font color="Green" title="E54">A21</font></span>: 
<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#K7" title="RELSET_1:func.7">.:</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#U2" title="MSUALG_1:sel.2">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:70_2"><span class="lab"><font color="Green" title="E46">A13</font></span></a>, <a class="txt" href="#E4:70_2"><span class="lab"><font color="Green" title="E47">A14</font></span></a>, <a class="ref" href="pua2mss1.html#D15" target="_self" title="PUA2MSS1:def.15">Def15</a></span>;<br/>
<a NAME="E14:70_2"/><span class="lab"><font color="Green" title="E55">A22</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:70_2"><span class="lab"><font color="Green" title="E46">A13</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>;<br/>
<a NAME="E15:70_2"/><span class="lab"><font color="Green" title="E56">A23</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:70_2"><span class="lab"><font color="Green" title="E46">A13</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>;<br/>
<a NAME="E16:70_2"/><span class="lab"><font color="Green" title="E57">A24</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">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#K7" title="RELSET_1:func.7">.:</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T148" title="RELAT_1:th.148">RELAT_1:148</a></span>;<br/>
<a NAME="E17:70_2"/><span class="lab"><font color="Green" title="E58">A25</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="txt" href="#E14:70_2"><span class="lab"><font color="Green" title="E55">A22</font></span></a>, <a class="txt" href="#E15:70_2"><span class="lab"><font color="Green" title="E56">A23</font></span></a></span>;<br/>
<a NAME="E18:70_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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pua2mss1.html#D11" target="_self" title="PUA2MSS1:def.11">Def11</a></span>;<br/>
<a NAME="E19:70_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A26</font></span>: 
 <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#K1" title="RELSET_1:func.1">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70_2"><span class="lab"><font color="Green" title="E47">A14</font></span></a>, <a class="ref" href="pua2mss1.html#D10" target="_self" title="PUA2MSS1:def.10">Def10</a></span>;<br/>
<a NAME="E20:70_2"/><span class="lab"><font color="Green" title="E60">A27</font></span>: 
<font color="Maroon" title="c4">Z</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">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="pboole.html#R6" title="PBOOLE:pred.6">=</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:70"><span class="lab"><font color="Green" title="E43">A4</font></span></a>, <a class="txt" href="#E6:70_2"><span class="lab"><font color="Green" title="E48">A15</font></span></a>, <a class="ref" href="relat_1.html#T79" title="RELAT_1:th.79">RELAT_1:79</a></span>;<br/>
<a NAME="E21:70_2"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:70_2"><span class="lab"><font color="Green" title="E59">A26</font></span></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></span>;<br/>
<span class="kw">hence </span><a NAME="E22:70_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="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<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#NK3" title="FUNCT_1:NK.3">*</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <span class="p4">(<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>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">Z</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <span class="p4">(<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>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:70_2"><span class="lab"><font color="Green" title="E53">A20</font></span></a>, <a class="txt" href="#E13:70_2"><span class="lab"><font color="Green" title="E54">A21</font></span></a>, <a class="txt" href="#E16:70_2"><span class="lab"><font color="Green" title="E57">A24</font></span></a>, <a class="txt" href="#E17:70_2"><span class="lab"><font color="Green" title="E58">A25</font></span></a>, <a class="txt" href="#E20:70_2"><span class="lab"><font color="Green" title="E60">A27</font></span></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">D</font> = <font color="Maroon" title="c5">D</font> as    <a href="pboole.html#M2" title="PBOOLE:mode.2">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="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">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="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">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="E14:70"/><span class="lab"><font color="Green" title="E45">A28</font></span>: 
 <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="struct_0.html#U4" title="STRUCT_0:sel.4">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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_3"><span class="kw">proof </span></a><div class="add">

<div><span class="kw">hereby </span> <span class="comment"><font color="Red">:: according to </font></span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c7">Q</font></span>
<div class="add"><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="struct_0.html#U4" title="STRUCT_0:sel.4">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> )</span><br/><span class="kw">assume </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="struct_0.html#U4" title="STRUCT_0:sel.4">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></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:70_3_1"/><span class="lab"><font color="Green" title="E46">A29</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:70_3_1"/><span class="lab"><font color="Green" title="E47">A30</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_5.html#T10" title="CARD_5:th.10">CARD_5:10</a></span>;<br/><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:70_3_1"><span class="lab"><font color="Green" title="E46">A29</font></span></a>, <a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>;<br/><a NAME="E6:70_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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:70"><span class="lab"><font color="Green" title="E41">A2</font></span></a></span>;<br/><a NAME="E7:70_3_1"/><span class="kw">then </span>
 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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70_3_1"><span class="lab"><font color="Green" title="E47">A30</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E8:70_3_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="struct_0.html#U4" title="STRUCT_0:sel.4">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> 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/>

<span class="kw">assume </span><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="struct_0.html#U4" title="STRUCT_0:sel.4">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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">then </span><span class="kw">reconsider </span><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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:70"><span class="lab"><font color="Green" title="E41">A2</font></span></a></span>;<br/>
<a NAME="E5:70_3"/><span class="kw">then </span>
<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/>
<span class="kw">hence </span><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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:70"><span class="lab"><font color="Green" title="E42">A3</font></span></a>, <a class="ref" href="card_5.html#T10" title="CARD_5:th.10">CARD_5:10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:70_4"/><span class="lab"><font color="Green" title="E46">A31</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E2:70_4"/><span class="lab"><font color="Green" title="E47">A32</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E3:70_4"/><span class="lab"><font color="Green" title="E48">A33</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_4"><span class="lab"><font color="Green" title="E46">A31</font></span></a>, <a class="txt" href="#E2:70_4"><span class="lab"><font color="Green" title="E47">A32</font></span></a>, <a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>;<br/><span class="kw">thus </span><a NAME="E5:70_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:70_4_1"/><span class="lab"><font color="Green" title="E49">A34</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:70_4_1"/><span class="lab"><font color="Green" title="E50">A35</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<a NAME="E5:70_4_1"/><span class="lab"><font color="Green" title="E51">A36</font></span>: 
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:70"><span class="lab"><font color="Green" title="E41">A2</font></span></a></span>;<br/>
<a NAME="E6:70_4_1"/><span class="lab"><font color="Green" title="E52">A37</font></span>: 
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:70"><span class="lab"><font color="Green" title="E41">A2</font></span></a></span>;<br/>
<a NAME="E7:70_4_1"/><span class="lab"><font color="Green" title="E53">A38</font></span>: 
 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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:70_4_1"><span class="lab"><font color="Green" title="E49">A34</font></span></a>, <a class="txt" href="#E5:70_4_1"><span class="lab"><font color="Green" title="E51">A36</font></span></a></span>;<br/>
<a NAME="E8:70_4_1"/>
 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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70_4_1"><span class="lab"><font color="Green" title="E50">A35</font></span></a>, <a class="txt" href="#E6:70_4_1"><span class="lab"><font color="Green" title="E52">A37</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:70_4_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:70_4"><span class="lab"><font color="Green" title="E48">A33</font></span></a>, <a class="txt" href="#E7:70_4_1"><span class="lab"><font color="Green" title="E53">A38</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><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="struct_0.html#U4" title="STRUCT_0:sel.4">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:70"><span class="lab"><font color="Green" title="E45">A28</font></span></a>, <a class="ref" href="pua2mss1.html#T15" target="_self" title="PUA2MSS1:th.15">Th15</a></span>;<br/>
<span class="kw">take </span><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> #); <span class="comment"><font color="Red">:: according to </font></span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="struct_0.html#U4" title="STRUCT_0:sel.4">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> 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/>

<span class="kw">take </span>
<font color="Maroon" title="c9">Q</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="E17:70"/>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T71" title="RELAT_1:th.71">RELAT_1:71</a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:70"/>
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pua2mss1.html#D12" target="_self" title="PUA2MSS1:def.12">Def12</a></span>; <span class="comment"><font color="Red">:: according to </font></span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#U4" title="MSUALG_1:sel.4">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/>

<span class="kw">thus </span><a NAME="E19:70"/>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#U4" title="MSUALG_1:sel.4">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/>

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">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/>

<span class="kw">reconsider </span><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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="pua2mss1.html#T9" target="_self" title="PUA2MSS1:th.9">Th9</a></span>;<br/>
<span class="kw">reconsider </span><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#K1" title="RELSET_1:func.1">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#K1" title="RELSET_1:func.1">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#K1" title="RELSET_1:func.1">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 class="kw">by</span> <span class="lab"><a class="ref" href="pua2mss1.html#T11" target="_self" title="PUA2MSS1:th.11">Th11</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">F</font> = the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">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="E22:70"/><span class="lab"><font color="Green" title="E46">A39</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:70"><span class="lab"><font color="Green" title="E42">A3</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E23:70"/><span class="lab"><font color="Green" title="E47">A40</font></span>: 
 <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="struct_0.html#U4" title="STRUCT_0:sel.4">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>;<br/>
<a NAME="E24:70"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A41</font></span>: 
 <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#U4" title="MSUALG_1:sel.4">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:70"><span class="lab"><font color="Green" title="E46">A39</font></span></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">F</font> = the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:70"><span class="lab"><font color="Green" title="E46">A39</font></span></a>, <a class="txt" href="#E23:70"><span class="lab"><font color="Green" title="E47">A40</font></span></a></span>;<br/>
<span class="kw">reconsider </span><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>  ;<br/>
<span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="ref" href="pua2mss1.html#T12" target="_self" title="PUA2MSS1:th.12">Th12</a></span>;<br/>
<a NAME="E28:70"/><span class="lab"><font color="Green" title="E49">A42</font></span>: 
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:70"><span class="lab"><font color="Green" title="E41">A2</font></span></a></span>;<br/>
<a NAME="E29:70"/>
 <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_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">assume </span><a NAME="E1:70_5"/>
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:70_5"/><span class="lab"><font color="Green" title="E50">A43</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:70_5"/><span class="lab"><font color="Green" title="E51">A44</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E5:70_5"/>
<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#U4" title="MSUALG_1:sel.4">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:70_5"><span class="lab"><font color="Green" title="E50">A43</font></span></a>, <a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>;<br/>
<a NAME="E6:70_5"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E8:70_5"/><span class="lab"><font color="Green" title="E52">A45</font></span>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">a</font>
 <span class="kw">and </span><br/><a NAME="E9:70_5"/><span class="lab"><font color="Green" title="E53">A46</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:70"><span class="lab"><font color="Green" title="E49">A42</font></span></a></span>;<br/>
<a NAME="E10:70_5"/>
<font color="Maroon" title="c20">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span>
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E12:70_5"/><span class="lab"><font color="Green" title="E54">A47</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E13:70_5"/><span class="lab"><font color="Green" title="E55">A48</font></span>: 
 <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#K1" title="RELSET_1:func.1">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:70"><span class="lab"><font color="Green" title="E44">A5</font></span></a></span>;<br/>
<a NAME="E14:70_5"/><span class="lab"><font color="Green" title="E56">A49</font></span>: 
<font color="Maroon" title="c21">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:70_5"><span class="lab"><font color="Green" title="E53">A46</font></span></a>, <a class="txt" href="#E12:70_5"><span class="lab"><font color="Green" title="E54">A47</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>;<br/>
<a NAME="E15:70_5"/><span class="lab"><font color="Green" title="E57">A50</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:70_5"><span class="lab"><font color="Green" title="E54">A47</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>;<br/>
<a NAME="E16:70_5"/><span class="lab"><font color="Green" title="E58">A51</font></span>: 
 <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>
 
;<br/>
<a NAME="E17:70_5"/><span class="lab"><font color="Green" title="E59">A52</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pua2mss1.html#D11" target="_self" title="PUA2MSS1:def.11">Def11</a></span>;<br/>
<a NAME="E18:70_5"/><span class="lab"><font color="Green" title="E60">A53</font></span>: 
<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#K1" title="RELSET_1:func.1">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:70_5"><span class="lab"><font color="Green" title="E55">A48</font></span></a>, <a class="txt" href="#E14:70_5"><span class="lab"><font color="Green" title="E56">A49</font></span></a>, <a class="txt" href="#E16:70_5"><span class="lab"><font color="Green" title="E58">A51</font></span></a></span>;<br/>
<a NAME="E19:70_5"/>
 <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#K1" title="RELSET_1:func.1">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E13:70_5"><span class="lab"><font color="Green" title="E55">A48</font></span></a>, <a class="txt" href="#E14:70_5"><span class="lab"><font color="Green" title="E56">A49</font></span></a>, <a class="txt" href="#E17:70_5"><span class="lab"><font color="Green" title="E59">A52</font></span></a>, <a class="ref" href="pua2mss1.html#D10" target="_self" title="PUA2MSS1:def.10">Def10</a></span>;<br/>
<a NAME="E20:70_5"/><span class="kw">then </span>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:70_5"><span class="lab"><font color="Green" title="E60">A53</font></span></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/>
<a NAME="E21:70_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A54</font></span>: 
<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>
 
;<br/>
<a NAME="E22:70_5"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:70_5"><span class="lab"><font color="Green" title="E50">A43</font></span></a>, <a class="txt" href="#E4:70_5"><span class="lab"><font color="Green" title="E51">A44</font></span></a>, <a class="txt" href="#E8:70_5"><span class="lab"><font color="Green" title="E52">A45</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>;<br/>
<span class="kw">hence </span><a NAME="E23:70_5"/>
<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 class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="txt" href="#E9:70_5"><span class="lab"><font color="Green" title="E53">A46</font></span></a>, <a class="txt" href="#E15:70_5"><span class="lab"><font color="Green" title="E57">A50</font></span></a>, <a class="txt" href="#E21:70_5"><span class="lab"><font color="Green" title="E61">A54</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:70"><span class="lab"><font color="Green" title="E48">A41</font></span></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></span>;<br/>
<a NAME="E31:70"/><span class="lab"><font color="Green" title="E50">A55</font></span>: 
<font color="Maroon" title="c17">RR</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c18">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c18">F</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:70_6"/>
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c18">F</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E3:70_6"/><span class="lab"><font color="Green" title="E51">A56</font></span>: 
<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_6"/>
<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/>
<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E6:70_6"/><span class="lab"><font color="Green" title="E52">A57</font></span>: 
<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#K1" title="RELSET_1:func.1">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>
 <span class="kw">and </span><br/><a NAME="E7:70_6"/><span class="lab"><font color="Green" title="E53">A58</font></span>: 
<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#K1" title="RELSET_1:func.1">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="E8:70_6"/>
<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/>
<span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E10:70_6"/><span class="lab"><font color="Green" title="E54">A59</font></span>: 
<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="E11:70_6"/>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pua2mss1.html#D11" target="_self" title="PUA2MSS1:def.11">Def11</a></span>;<br/>
<a NAME="E12:70_6"/><span class="kw">then </span>
 <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#K1" title="RELSET_1:func.1">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E7:70_6"><span class="lab"><font color="Green" title="E53">A58</font></span></a>, <a class="txt" href="#E10:70_6"><span class="lab"><font color="Green" title="E54">A59</font></span></a>, <a class="ref" href="pua2mss1.html#D10" target="_self" title="PUA2MSS1:def.10">Def10</a></span>;<br/>
<a NAME="E13:70_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A60</font></span>: 
<font color="Maroon" title="c21">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:70_6"><span class="lab"><font color="Green" title="E52">A57</font></span></a>, <a class="txt" href="#E10:70_6"><span class="lab"><font color="Green" title="E54">A59</font></span></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/>
<a NAME="E14:70_6"/><span class="lab"><font color="Green" title="E56">A61</font></span>: 
<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="struct_0.html#U4" title="STRUCT_0:sel.4">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:70"><span class="lab"><font color="Green" title="E44">A5</font></span></a>, <a class="txt" href="#E7:70_6"><span class="lab"><font color="Green" title="E53">A58</font></span></a>, <a class="txt" href="#E10:70_6"><span class="lab"><font color="Green" title="E54">A59</font></span></a></span>;<br/>
<a NAME="E15:70_6"/><span class="lab"><font color="Green" title="E57">A62</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>;<br/>
<a NAME="E16:70_6"/><span class="lab"><font color="Green" title="E58">A63</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>;<br/>
<a NAME="E17:70_6"/><span class="lab"><font color="Green" title="E59">A64</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:70_6"><span class="lab"><font color="Green" title="E56">A61</font></span></a>, <a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>;<br/>
<a NAME="E18:70_6"/><span class="lab"><font color="Green" title="E60">A65</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:70"><span class="lab"><font color="Green" title="E49">A42</font></span></a>, <a class="txt" href="#E14:70_6"><span class="lab"><font color="Green" title="E56">A61</font></span></a>, <a class="txt" href="#E15:70_6"><span class="lab"><font color="Green" title="E57">A62</font></span></a></span>;<br/>
<a NAME="E19:70_6"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="txt" href="#E3:70_6"><span class="lab"><font color="Green" title="E51">A56</font></span></a>, <a class="txt" href="#E10:70_6"><span class="lab"><font color="Green" title="E54">A59</font></span></a>, <a class="txt" href="#E13:70_6"><span class="lab"><font color="Green" title="E55">A60</font></span></a>, <a class="txt" href="#E14:70_6"><span class="lab"><font color="Green" title="E56">A61</font></span></a>, <a class="txt" href="#E15:70_6"><span class="lab"><font color="Green" title="E57">A62</font></span></a>, <a class="txt" href="#E16:70_6"><span class="lab"><font color="Green" title="E58">A63</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E20:70_6"/>
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c18">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:70_6"><span class="lab"><font color="Green" title="E59">A64</font></span></a>, <a class="txt" href="#E18:70_6"><span class="lab"><font color="Green" title="E60">A65</font></span></a>, <a class="ref" href="funct_1.html#T73" title="FUNCT_1:th.73">FUNCT_1:73</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E32:70"/>
<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_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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> ; <span class="comment"><font color="Red">:: according to </font></span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K1" title="RELAT_1:func.1">proj1</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#K1" title="RELAT_1:func.1">proj1</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/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:70_7"/><span class="lab"><font color="Green" title="E51">A66</font></span>: 
<font color="Maroon" title="c19">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c18">F</font>
 <span class="kw">and </span><br/><a NAME="E2:70_7"/><span class="lab"><font color="Green" title="E52">A67</font></span>: 
<font color="Maroon" title="c20">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c18">F</font>
 <span class="kw">and </span><br/><a NAME="E3:70_7"/><span class="lab"><font color="Green" title="E53">A68</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E5:70_7"/><span class="lab"><font color="Green" title="E54">A69</font></span>: 
<font color="Maroon" title="c19">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">a1</font>
 <span class="kw">and </span><br/><a NAME="E6:70_7"/><span class="lab"><font color="Green" title="E55">A70</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:70"><span class="lab"><font color="Green" title="E48">A41</font></span></a>, <a class="txt" href="#E28:70"><span class="lab"><font color="Green" title="E49">A42</font></span></a>, <a class="txt" href="#E1:70_7"><span class="lab"><font color="Green" title="E51">A66</font></span></a></span>;<br/>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E8:70_7"/><span class="lab"><font color="Green" title="E56">A71</font></span>: 
<font color="Maroon" title="c20">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">a2</font>
 <span class="kw">and </span><br/><a NAME="E9:70_7"/><span class="lab"><font color="Green" title="E57">A72</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:70"><span class="lab"><font color="Green" title="E48">A41</font></span></a>, <a class="txt" href="#E28:70"><span class="lab"><font color="Green" title="E49">A42</font></span></a>, <a class="txt" href="#E2:70_7"><span class="lab"><font color="Green" title="E52">A67</font></span></a></span>;<br/>
<a NAME="E10:70_7"/>
<font color="Maroon" title="c21">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span>
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E12:70_7"/><span class="lab"><font color="Green" title="E58">A73</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E13:70_7"/><span class="lab"><font color="Green" title="E59">A74</font></span>: 
 <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#K1" title="RELSET_1:func.1">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:70"><span class="lab"><font color="Green" title="E44">A5</font></span></a></span>;<br/>
<a NAME="E14:70_7"/>
<font color="Maroon" title="c22">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span>
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E16:70_7"/><span class="lab"><font color="Green" title="E60">A75</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E17:70_7"/><span class="lab"><font color="Green" title="E61">A76</font></span>: 
 <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#K1" title="RELSET_1:func.1">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:70"><span class="lab"><font color="Green" title="E44">A5</font></span></a></span>;<br/>
<a NAME="E18:70_7"/><span class="lab"><font color="Green" title="E62">A77</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_7"><span class="lab"><font color="Green" title="E51">A66</font></span></a>, <a class="txt" href="#E5:70_7"><span class="lab"><font color="Green" title="E54">A69</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>;<br/>
<a NAME="E19:70_7"/><span class="lab"><font color="Green" title="E63">A78</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:70_7"><span class="lab"><font color="Green" title="E52">A67</font></span></a>, <a class="txt" href="#E3:70_7"><span class="lab"><font color="Green" title="E53">A68</font></span></a>, <a class="txt" href="#E8:70_7"><span class="lab"><font color="Green" title="E56">A71</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>;<br/>
<a NAME="E20:70_7"/><span class="lab"><font color="Green" title="E64">A79</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:70_7"><span class="lab"><font color="Green" title="E58">A73</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>;<br/>
<a NAME="E21:70_7"/><span class="lab"><font color="Green" title="E65">A80</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:70_7"><span class="lab"><font color="Green" title="E60">A75</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>;<br/>
<a NAME="E22:70_7"/><span class="lab"><font color="Green" title="E66">A81</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="txt" href="#E6:70_7"><span class="lab"><font color="Green" title="E55">A70</font></span></a>, <a class="txt" href="#E18:70_7"><span class="lab"><font color="Green" title="E62">A77</font></span></a>, <a class="txt" href="#E20:70_7"><span class="lab"><font color="Green" title="E64">A79</font></span></a></span>;<br/>
<a NAME="E23:70_7"/><span class="lab"><font color="Green" title="E67">A82</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="txt" href="#E9:70_7"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="txt" href="#E19:70_7"><span class="lab"><font color="Green" title="E63">A78</font></span></a>, <a class="txt" href="#E21:70_7"><span class="lab"><font color="Green" title="E65">A80</font></span></a></span>;<br/>
<a NAME="E24:70_7"/><span class="lab"><font color="Green" title="E68">A83</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pua2mss1.html#D11" target="_self" title="PUA2MSS1:def.11">Def11</a></span>;<br/>
<a NAME="E25:70_7"/><span class="lab"><font color="Green" title="E69">A84</font></span>: 
<font color="Maroon" title="c11">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">o1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:70_7"><span class="lab"><font color="Green" title="E55">A70</font></span></a>, <a class="txt" href="#E12:70_7"><span class="lab"><font color="Green" title="E58">A73</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>;<br/>
<a NAME="E26:70_7"/><span class="lab"><font color="Green" title="E70">A85</font></span>: 
<font color="Maroon" title="c11">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">o2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:70_7"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="txt" href="#E16:70_7"><span class="lab"><font color="Green" title="E60">A75</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>;<br/>
<a NAME="E27:70_7"/><span class="lab"><font color="Green" title="E71">A86</font></span>: 
 <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#K1" title="RELSET_1:func.1">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E13:70_7"><span class="lab"><font color="Green" title="E59">A74</font></span></a>, <a class="txt" href="#E24:70_7"><span class="lab"><font color="Green" title="E68">A83</font></span></a>, <a class="txt" href="#E25:70_7"><span class="lab"><font color="Green" title="E69">A84</font></span></a>, <a class="ref" href="pua2mss1.html#D10" target="_self" title="PUA2MSS1:def.10">Def10</a></span>;<br/>
<a NAME="E28:70_7"/><span class="lab"><font color="Green" title="E72">A87</font></span>: 
 <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#K1" title="RELSET_1:func.1">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E17:70_7"><span class="lab"><font color="Green" title="E61">A76</font></span></a>, <a class="txt" href="#E24:70_7"><span class="lab"><font color="Green" title="E68">A83</font></span></a>, <a class="txt" href="#E26:70_7"><span class="lab"><font color="Green" title="E70">A85</font></span></a>, <a class="ref" href="pua2mss1.html#D10" target="_self" title="PUA2MSS1:def.10">Def10</a></span>;<br/>
<a NAME="E29:70_7"/>
 <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#K1" title="RELSET_1:func.1">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:70_7"><span class="lab"><font color="Green" title="E71">A86</font></span></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></span>;<br/>
<span class="kw">hence </span><a NAME="E30:70_7"/>
<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 class="kw">by</span> <span class="lab"><a class="txt" href="#E5:70_7"><span class="lab"><font color="Green" title="E54">A69</font></span></a>, <a class="txt" href="#E8:70_7"><span class="lab"><font color="Green" title="E56">A71</font></span></a>, <a class="txt" href="#E12:70_7"><span class="lab"><font color="Green" title="E58">A73</font></span></a>, <a class="txt" href="#E16:70_7"><span class="lab"><font color="Green" title="E60">A75</font></span></a>, <a class="txt" href="#E22:70_7"><span class="lab"><font color="Green" title="E66">A81</font></span></a>, <a class="txt" href="#E23:70_7"><span class="lab"><font color="Green" title="E67">A82</font></span></a>, <a class="txt" href="#E25:70_7"><span class="lab"><font color="Green" title="E69">A84</font></span></a>, <a class="txt" href="#E26:70_7"><span class="lab"><font color="Green" title="E70">A85</font></span></a>, <a class="txt" href="#E28:70_7"><span class="lab"><font color="Green" title="E72">A87</font></span></a>, <a class="ref" href="pua2mss1.html#T2" target="_self" title="PUA2MSS1:th.2">Th2</a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E33:70"/>
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E31:70"><span class="lab"><font color="Green" title="E50">A55</font></span></a>, <a class="ref" href="pua2mss1.html#T16" target="_self" title="PUA2MSS1:th.16">Th16</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
