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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="unialg_1.html#V6">partial</a> <a href="unialg_1.html#V8">non-empty</a>  <a href="unialg_1.html#L1">UAStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="pua2mss1.html#M3" target="_self">a_partition</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E1:71"/>
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K4" target="_self">id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="pua2mss1.html#D15" target="_self">Def15</a>, <a class="ref" href="relat_1.html#T71">RELAT_1:71</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> =  <a href="pua2mss1.html#K4" target="_self">id</a> <font color="Maroon">c<sub>2</sub></font> as    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="msualg_1.html#NM2">OperSymbol</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>) <b>-&gt; </b>   <a href="relset_1.html#M2">Relation</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K3">*</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> = <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" target="_self">Den</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="pua2mss1.html#K12" target="_self">`1</a> </span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="pua2mss1.html#K13" target="_self">`2</a> </span>)</span></span>)</span>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E4:71"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> holds <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="pboole.html#S5">PBOOLE:sch 5</a>();<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="pua2mss1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="msualg_1.html#NM2">OperSymbol</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> : <font color="Olive">b<sub>1</sub></font> <a href="pua2mss1.html#K12" target="_self">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> </span>}</span> ;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="pboole.html#M1">ManySortedSet</a> of  <a href="finseq_1.html#K4">dom</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:71"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="pua2mss1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="pboole.html#S5">PBOOLE:sch 5</a>();<br/></i>
<a NAME="E7:71"/><i><font color="Green">E43</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E8:71"/><i><font color="Green">E44</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> where B is   <a href="pua2mss1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font>, B is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>2</sub></font> <a href="finseq_2.html#K3">*</a>  :  <a href="card_3.html#K4">product</a> <font color="Olive">b<sub>2</sub></font> <a href="subset_1.html#R2">meets</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> </span>}</span>  )
 
<b>by </b><i><a class="ref" href="pua2mss1.html#D15" target="_self">Def15</a></i>;<br/>
<a NAME="E9:71"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="relat_1.html#V2">non-empty</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D15">FUNCT_1:def 15</a><br/>

<b>assume </b><a NAME="E1:71_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="pua2mss1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as    <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K3">*</a>  ;<br/>
<a NAME="E5:71_1"/>
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E7:71_1"/><i><font color="Green">E45</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="ref" href="pua2mss1.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E8:71_1"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E9:71_1"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> is   <a href="finseq_1.html#NM1">FinSequence</a>
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E7:71_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="finseq_1.html#D2">FINSEQ_1:def 2</a></i>;<br/>
<a NAME="E10:71_1"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> is    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E7:71_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>2</sub></font> <a href="finseq_2.html#K3">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E12:71_1"/>
 <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>12</sub></font> <a href="subset_1.html#R2">meets</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E7:71_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E13:71_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="msualg_1.html#NM2">OperSymbol</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> : <font color="Olive">b<sub>1</sub></font> <a href="pua2mss1.html#K12" target="_self">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> </span>}</span>  )
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E6:71"><i><font color="Green">E42</font></i></a>, <a class="txt" href="pua2mss1.html#E8:71"><i><font color="Green">E44</font></i></a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E14:71_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>
 
;<br/>
<b>hence </b><a NAME="E15:71_1"/>
not <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as  <a href="relat_1.html#V2">non-empty</a> <a href="funct_1.html#NM1">Function</a> ;<br/>
<a NAME="E11:71"/>
<font color="Maroon">c<sub>5</sub></font> is    <a href="pboole.html#M3">ManySortedFunction</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D18">PBOOLE:def 18</a><br/>

<b>assume </b><a NAME="E1:71_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="pua2mss1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>10</sub></font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>2</sub></font> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E3:71_2"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span> &amp;  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#R2">meets</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="pua2mss1.html#E8:71"><i><font color="Green">E44</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="msualg_1.html#NM2">OperSymbol</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="txt" href="pua2mss1.html#E1:71_2"><i><font color="Green">E45</font></i></a></i>;<br/>
<a NAME="E5:71_2"/><i><font color="Green">E47</font></i>: 
(  <a href="relset_1.html#K5">rng</a> the <a href="msualg_1.html#U3">ResultSort</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> the <a href="msualg_1.html#U2">Arity</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:71_2"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
;<br/>
<a NAME="E7:71_2"/><b>then </b>
( the <a href="msualg_1.html#U2">Arity</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> )
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E8:71"><i><font color="Green">E44</font></i></a>, <a class="txt" href="pua2mss1.html#E3:71_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="pua2mss1.html#D15" target="_self">Def15</a>, <a class="ref" href="pboole.html#D19">PBOOLE:def 19</a>, <a class="ref" href="relat_1.html#T79">RELAT_1:79</a></i>;<br/>
<a NAME="E8:71_2"/><b>then </b><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <span class="p2">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E1:71_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="pua2mss1.html#E5:71_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>
<a NAME="E9:71_2"/><i><font color="Green">E49</font></i>: 
( <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U3">ResultSort</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="pua2mss1.html#K13" target="_self">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="pua2mss1.html#K12" target="_self">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E3:71_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="pua2mss1.html#D15" target="_self">Def15</a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E10:71_2"/><b>then </b><i><font color="Green">E50</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E4:71"><i><font color="Green">E41</font></i></a>, <a class="ref" href="relat_1.html#T148">RELAT_1:148</a></i>;<br/>
<a NAME="E11:71_2"/>
 <a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="pua2mss1.html#R2" target="_self">is_exactly_partitable_wrt</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="pua2mss1.html#D11" target="_self">Def11</a></i>;<br/>
<a NAME="E12:71_2"/><b>then </b>
 <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E3:71_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="pua2mss1.html#D10" target="_self">Def10</a></i>;<br/>
<a NAME="E13:71_2"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U3">ResultSort</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>10</sub></font> )
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E8:71"><i><font color="Green">E44</font></i></a>, <a class="txt" href="pua2mss1.html#E5:71_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="relat_1.html#T79">RELAT_1:79</a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<b>hence </b><a NAME="E14:71_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> is    <a href="relset_1.html#M2">Relation</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <span class="p2">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <span class="p2">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="pua2mss1.html#E8:71_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="pua2mss1.html#E9:71_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="pua2mss1.html#E10:71_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="pboole.html#M3">ManySortedFunction</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> ;<br/>
<a NAME="E13:71"/><i><font color="Green">E45</font></i>: 
 <a href="card_3.html#K3">Union</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_3"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>,<a class="ref" href="tarski.html#D3">TARSKI:def 3</a>
<div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:71_3_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K3">Union</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:71_3_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="ref" href="card_5.html#T10">CARD_5:10</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="pua2mss1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="pua2mss1.html#E3:71_3_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E5:71_3_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="msualg_1.html#NM2">OperSymbol</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> : <font color="Olive">b<sub>1</sub></font> <a href="pua2mss1.html#K12" target="_self">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="pua2mss1.html#E6:71"><i><font color="Green">E42</font></i></a></i>;<br/><a NAME="E6:71_3_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> st <br/>( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="pua2mss1.html#K12" target="_self">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="txt" href="pua2mss1.html#E3:71_3_1"><i><font color="Green">E46</font></i></a></i>;<br/><b>hence </b><a NAME="E7:71_3_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:71_3"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="msualg_1.html#NM2">OperSymbol</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> ;<br/>
<a NAME="E4:71_3"/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="pua2mss1.html#K12" target="_self">`1</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="msualg_1.html#NM2">OperSymbol</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> : <font color="Olive">b<sub>1</sub></font> <a href="pua2mss1.html#K12" target="_self">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="pua2mss1.html#K12" target="_self">`1</a>  </span>}</span> 
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E6:71"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E5:71_3"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="pua2mss1.html#K12" target="_self">`1</a> </span>)</span>
 
;<br/>
<b>hence </b><a NAME="E6:71_3"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K3">Union</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="pua2mss1.html#E7:71"><i><font color="Green">E43</font></i></a>, <a class="ref" href="card_5.html#T10">CARD_5:10</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:71_4"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>10</sub></font> )
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="pua2mss1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><b>thus </b><a NAME="E3:71_4"/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:71_4_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:71_4_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E4:71_4_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="msualg_1.html#NM2">OperSymbol</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> : <font color="Olive">b<sub>1</sub></font> <a href="pua2mss1.html#K12" target="_self">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> </span>}</span>  &amp; <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="msualg_1.html#NM2">OperSymbol</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> : <font color="Olive">b<sub>1</sub></font> <a href="pua2mss1.html#K12" target="_self">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> </span>}</span>  )
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E6:71"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E5:71_4_1"/><b>then </b>
( ex <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> st <br/>( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="pua2mss1.html#K12" target="_self">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> ) &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> st <br/>( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="pua2mss1.html#K12" target="_self">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> ) )
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E3:71_4_1"><i><font color="Green">E47</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:71_4_1"/>
contradiction
 <b>by </b><i><a class="txt" href="pua2mss1.html#E1:71_4"><i><font color="Green">E46</font></i></a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> as    <a href="pua2mss1.html#M2" target="_self">IndexedPartition</a> of the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="txt" href="pua2mss1.html#E13:71"><i><font color="Green">E45</font></i></a>, <a class="ref" href="pua2mss1.html#T15" target="_self">Th15</a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>10</sub></font> =  <a href="msualg_1.html#G3">MSAlgebra</a>(# <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>8</sub></font> #); <i><font color="Red">:: according to </font></i><a class="ref" href="pua2mss1.html#D17" target="_self">PUA2MSS1:def 17</a><br/>

<b>take </b>
<font color="Maroon">c<sub>9</sub></font>
;<br/>

<a NAME="E16:71"/>
 <a href="pua2mss1.html#K4" target="_self">id</a> <font color="Maroon">c<sub>2</sub></font> is    <a href="pua2mss1.html#M4" target="_self">IndexedPartition</a> of <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_5"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:71_5"/>
 <a href="pua2mss1.html#K3" target="_self">rng</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K4" target="_self">id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> is    <a href="pua2mss1.html#M3" target="_self">a_partition</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#T71">RELAT_1:71</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pua2mss1.html#D12" target="_self">PUA2MSS1:def 12</a><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E17:71"/>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>10</sub></font> is    <a href="pua2mss1.html#M4" target="_self">IndexedPartition</a> of <font color="Maroon">c<sub>1</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="pua2mss1.html#D16" target="_self">PUA2MSS1:def 16</a><br/>

<b>thus </b><a NAME="E18:71"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="pua2mss1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>2</sub></font> <a href="finseq_2.html#K3">*</a>  : verum </span>}</span>  as    <a href="eqrel_1.html#M1">a_partition</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K3">*</a>  <b>by </b><i><a class="ref" href="pua2mss1.html#T9" target="_self">Th9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p4">(<span class="default"><a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>12</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#R2">meets</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> </span>}</span>  as    <a href="eqrel_1.html#M1">a_partition</a> of  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="pua2mss1.html#T11" target="_self">Th11</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>14</sub></font> = the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>10</sub></font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>;<br/>
<a NAME="E21:71"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="pua2mss1.html#K3" target="_self">rng</a> <font color="Maroon">c<sub>9</sub></font> &amp;  <a href="pua2mss1.html#K3" target="_self">rng</a> <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E7:71"><i><font color="Green">E43</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E22:71"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E23:71"/><b>then </b><i><font color="Green">E47</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>10</sub></font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="finseq_1.html#K4">dom</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E21:71"><i><font color="Green">E46</font></i></a>, <a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>10</sub></font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> as  non <a href="xboole_0.html#V1">empty</a> <a href="funct_1.html#NM1">Function</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="pua2mss1.html#E21:71"><i><font color="Green">E46</font></i></a>, <a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>13</sub></font> : verum </span>}</span>  as    <a href="eqrel_1.html#M1">a_partition</a> of  <a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="pua2mss1.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E27:71"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="msualg_1.html#NM2">OperSymbol</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> : <font color="Olive">b<sub>1</sub></font> <a href="pua2mss1.html#K12" target="_self">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E6:71"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E28:71"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>17</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:71_6"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>19</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:71_6"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:71_6"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E3:71_6"><i><font color="Green">E49</font></i></a>, <a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<a NAME="E5:71_6"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>20</sub></font> being   <a href="msualg_1.html#NM2">OperSymbol</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E7:71_6"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="pua2mss1.html#K12" target="_self">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="txt" href="pua2mss1.html#E27:71"><i><font color="Green">E48</font></i></a></i>;<br/>
<a NAME="E8:71_6"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>21</sub></font> being   <a href="pua2mss1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>22</sub></font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>2</sub></font> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E10:71_6"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></font></span><a href="tarski.html#K4">]</a></span> &amp;  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>22</sub></font> <a href="subset_1.html#R2">meets</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="pua2mss1.html#E8:71"><i><font color="Green">E44</font></i></a></i>;<br/>
<a NAME="E11:71_6"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="pua2mss1.html#K13" target="_self">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp;  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> &amp;  <a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="pua2mss1.html#R2" target="_self">is_exactly_partitable_wrt</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E7:71_6"><i><font color="Green">E50</font></i></a>, <a class="txt" href="pua2mss1.html#E10:71_6"><i><font color="Green">E51</font></i></a>, <a class="ref" href="pua2mss1.html#D11" target="_self">Def11</a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E12:71_6"/><b>then </b>
( <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> &amp;  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>22</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E10:71_6"><i><font color="Green">E51</font></i></a>, <a class="ref" href="pua2mss1.html#D10" target="_self">Def10</a></i>;<br/>
<a NAME="E13:71_6"/><b>then </b>
 <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E14:71_6"/><b>then </b>
( <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> )
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E3:71_6"><i><font color="Green">E49</font></i></a>, <a class="txt" href="pua2mss1.html#E7:71_6"><i><font color="Green">E50</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<b>hence </b><a NAME="E15:71_6"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="pua2mss1.html#E4:71"><i><font color="Green">E41</font></i></a>, <a class="txt" href="pua2mss1.html#E7:71_6"><i><font color="Green">E50</font></i></a>, <a class="txt" href="pua2mss1.html#E11:71_6"><i><font color="Green">E52</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>15</sub></font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> <b>by </b><i><a class="txt" href="pua2mss1.html#E23:71"><i><font color="Green">E47</font></i></a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<a NAME="E30:71"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>18</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>19</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:71_7"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>20</sub></font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>13</sub></font><b> such that </b><br/><a NAME="E3:71_7"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>20</sub></font>
 ;<br/>
<a NAME="E4:71_7"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font>
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>21</sub></font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>12</sub></font><b> such that </b><br/><a NAME="E6:71_7"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="subset_1.html#R2">meets</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> )
 ;<br/>
<a NAME="E7:71_7"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>22</sub></font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>2</sub></font> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E9:71_7"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>22</sub></font>
 ;<br/>
<a NAME="E10:71_7"/>
 <a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="pua2mss1.html#R2" target="_self">is_exactly_partitable_wrt</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="pua2mss1.html#D11" target="_self">Def11</a></i>;<br/>
<a NAME="E11:71_7"/><b>then </b>
 <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>22</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E6:71_7"><i><font color="Green">E51</font></i></a>, <a class="txt" href="pua2mss1.html#E9:71_7"><i><font color="Green">E52</font></i></a>, <a class="ref" href="pua2mss1.html#D10" target="_self">Def10</a></i>;<br/>
<a NAME="E12:71_7"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E6:71_7"><i><font color="Green">E51</font></i></a>, <a class="txt" href="pua2mss1.html#E9:71_7"><i><font color="Green">E52</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E13:71_7"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>22</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>22</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>22</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> )
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E8:71"><i><font color="Green">E44</font></i></a>, <a class="txt" href="pua2mss1.html#E6:71_7"><i><font color="Green">E51</font></i></a>, <a class="txt" href="pua2mss1.html#E9:71_7"><i><font color="Green">E52</font></i></a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E14:71_7"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>22</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>8</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>22</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>22</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> )
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E4:71"><i><font color="Green">E41</font></i></a>, <a class="txt" href="pua2mss1.html#E27:71"><i><font color="Green">E48</font></i></a>, <a class="txt" href="pua2mss1.html#E3:71_7"><i><font color="Green">E50</font></i></a>, <a class="txt" href="pua2mss1.html#E9:71_7"><i><font color="Green">E52</font></i></a>, <a class="txt" href="pua2mss1.html#E12:71_7"><i><font color="Green">E53</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>hence </b><a NAME="E15:71_7"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#T73">FUNCT_1:73</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E31:71"/>
<font color="Maroon">c<sub>18</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>19</sub></font>, <font color="Maroon">c<sub>20</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/>

<b>assume </b><a NAME="E1:71_8"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> )
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>21</sub></font> being   <a href="msualg_1.html#NM2">OperSymbol</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:71_8"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="pua2mss1.html#K12" target="_self">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="txt" href="pua2mss1.html#E23:71"><i><font color="Green">E47</font></i></a>, <a class="txt" href="pua2mss1.html#E27:71"><i><font color="Green">E48</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>22</sub></font> being   <a href="msualg_1.html#NM2">OperSymbol</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E5:71_8"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="pua2mss1.html#K12" target="_self">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="txt" href="pua2mss1.html#E23:71"><i><font color="Green">E47</font></i></a>, <a class="txt" href="pua2mss1.html#E27:71"><i><font color="Green">E48</font></i></a>, <a class="txt" href="pua2mss1.html#E1:71_8"><i><font color="Green">E50</font></i></a></i>;<br/>
<a NAME="E6:71_8"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>23</sub></font> being   <a href="pua2mss1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>24</sub></font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>2</sub></font> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E8:71_8"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font></span><a href="tarski.html#K4">]</a></span> &amp;  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>24</sub></font> <a href="subset_1.html#R2">meets</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="pua2mss1.html#E8:71"><i><font color="Green">E44</font></i></a></i>;<br/>
<a NAME="E9:71_8"/>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>25</sub></font> being   <a href="pua2mss1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>26</sub></font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>2</sub></font> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E11:71_8"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="tarski.html#K4">]</a></span> &amp;  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>26</sub></font> <a href="subset_1.html#R2">meets</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="pua2mss1.html#E8:71"><i><font color="Green">E44</font></i></a></i>;<br/>
<a NAME="E12:71_8"/>
( <font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="pua2mss1.html#K13" target="_self">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="pua2mss1.html#K13" target="_self">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> )
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E1:71_8"><i><font color="Green">E50</font></i></a>, <a class="txt" href="pua2mss1.html#E3:71_8"><i><font color="Green">E51</font></i></a>, <a class="txt" href="pua2mss1.html#E5:71_8"><i><font color="Green">E52</font></i></a>, <a class="txt" href="pua2mss1.html#E8:71_8"><i><font color="Green">E53</font></i></a>, <a class="txt" href="pua2mss1.html#E11:71_8"><i><font color="Green">E54</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E13:71_8"/><b>then </b><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>26</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E4:71"><i><font color="Green">E41</font></i></a>, <a class="txt" href="pua2mss1.html#E3:71_8"><i><font color="Green">E51</font></i></a>, <a class="txt" href="pua2mss1.html#E5:71_8"><i><font color="Green">E52</font></i></a></i>;<br/>
<a NAME="E14:71_8"/><i><font color="Green">E56</font></i>: 
(  <a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="pua2mss1.html#R2" target="_self">is_exactly_partitable_wrt</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font> )
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E3:71_8"><i><font color="Green">E51</font></i></a>, <a class="txt" href="pua2mss1.html#E5:71_8"><i><font color="Green">E52</font></i></a>, <a class="txt" href="pua2mss1.html#E8:71_8"><i><font color="Green">E53</font></i></a>, <a class="txt" href="pua2mss1.html#E11:71_8"><i><font color="Green">E54</font></i></a>, <a class="ref" href="pua2mss1.html#D11" target="_self">Def11</a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E15:71_8"/><b>then </b>
(  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>24</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>26</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E8:71_8"><i><font color="Green">E53</font></i></a>, <a class="txt" href="pua2mss1.html#E11:71_8"><i><font color="Green">E54</font></i></a>, <a class="ref" href="pua2mss1.html#D10" target="_self">Def10</a></i>;<br/>
<a NAME="E16:71_8"/><b>then </b>
(  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span>)</span> &amp;  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>26</sub></font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<b>hence </b><a NAME="E17:71_8"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="pua2mss1.html#E3:71_8"><i><font color="Green">E51</font></i></a>, <a class="txt" href="pua2mss1.html#E5:71_8"><i><font color="Green">E52</font></i></a>, <a class="txt" href="pua2mss1.html#E8:71_8"><i><font color="Green">E53</font></i></a>, <a class="txt" href="pua2mss1.html#E11:71_8"><i><font color="Green">E54</font></i></a>, <a class="txt" href="pua2mss1.html#E13:71_8"><i><font color="Green">E55</font></i></a>, <a class="txt" href="pua2mss1.html#E14:71_8"><i><font color="Green">E56</font></i></a>, <a class="ref" href="pua2mss1.html#T2" target="_self">Th2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E32:71"/>
the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>10</sub></font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> is    <a href="pua2mss1.html#M2" target="_self">IndexedPartition</a> of  <a href="pua2mss1.html#K2" target="_self">Den</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="pua2mss1.html#E30:71"><i><font color="Green">E49</font></i></a>, <a class="ref" href="pua2mss1.html#T16" target="_self">Th16</a></i>;<br/>


</div>
