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

<a NAME="E1:25_1"/><i><font color="Green">E13</font></i>: 
 <a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#G3">MSAlgebra</a>(# <span class="p1">(<span class="default"><a href="msafree.html#K8">FreeSort</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>,<span class="p1">(<span class="default"><a href="msafree.html#K10">FreeOper</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> #)
 
<b>by </b><i><a class="ref" href="msafree.html#D16">MSAFREE:def 16</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>1</sub></font> = <font color="Maroon">F<sub>4</sub></font>(), <font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">F<sub>5</sub></font>() as    <a href="pboole.html#M3">ManySortedFunction</a> of  <a href="msafree.html#K8">FreeSort</a> <font color="Maroon">F<sub>2</sub></font>(),the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>3</sub></font>() ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means for <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>() st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree.html#K12">FreeGen</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">F<sub>2</sub></font>() holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font>,<font color="Maroon">a<sub>1</sub></font>];<br/>
<a NAME="E3:25_1"/><i><font color="Green">E14</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K3">Union</a> <span class="p1">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K3">Union</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>3</sub></font>() &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:25_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K3">Union</a> <span class="p1">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:25_1_1"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</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>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="txt" href="msafree1.html#E3:25_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>;<br/>

<a NAME="E5:25_1_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree.html#K12">FreeGen</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">F<sub>2</sub></font>()
 
<b>by </b><i><a class="txt" href="msafree1.html#E3:25_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="msafree.html#D18">MSAFREE:def 18</a></i>;<br/>
<a NAME="E6:25_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><a href="msafree.html#K8">FreeSort</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
 
;<br/>
<a NAME="E7:25_1_1"/><b>then </b><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="msafree1.html#E5:25_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<a NAME="E8:25_1_1"/>
 <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>3</sub></font>() <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>hence </b><a NAME="E9:25_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K3">Union</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>3</sub></font>()
 <b>by </b><i><a class="txt" href="msafree1.html#E7:25_1_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="card_5.html#T10">CARD_5:10</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<b>assume </b><a NAME="E10:25_1_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree.html#K12">FreeGen</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">F<sub>2</sub></font>()
 ;<br/>

<a NAME="E11:25_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K8">FreeSort</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K8">FreeSort</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="msafree1.html#E5:25_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E12:25_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="msafree.html#K8">FreeSort</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#R2">meets</a> <span class="p1">(<span class="default"><a href="msafree.html#K8">FreeSort</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E13:25_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="msafree.html#T13">MSAFREE:13</a></i>;<br/>
<b>hence </b><a NAME="E14:25_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>]
 <b>by </b><i><a class="txt" href="msafree1.html#E2:25"><i><font color="Green">E10</font></i></a>, <a class="txt" href="msafree1.html#E10:25_1_1"><i><font color="Green">E18</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="card_3.html#K3">Union</a> <span class="p1">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>, <a href="card_3.html#K3">Union</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>3</sub></font>()<b> such that </b><br/><a NAME="E5:25_1"/><i><font color="Green">E15</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K3">Union</a> <span class="p1">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>(<a class="txt" href="msafree1.html#E3:25_1"><i><font color="Green">E14</font></i></a>);<br/></i>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:25_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <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="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">F<sub>1</sub></font>(), B is    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">F<sub>3</sub></font>() : verum </span>}</span> 
 ;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means for <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">F<sub>3</sub></font>() st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><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> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">F<sub>3</sub></font>()</span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font>;<br/>
<a NAME="E8:25_1"/><i><font color="Green">E17</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K3">Union</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>3</sub></font>() &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:25_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">c<sub>7</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">F<sub>3</sub></font>()<b> such that </b><br/><a NAME="E3:25_1_2"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>5</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>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="msafree1.html#E7:25_1"><i><font color="Green">E16</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">F<sub>3</sub></font>()</span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> as    <a href="hidden.html#M1">set</a>  ;<br/>
<b>take </b>
<font color="Maroon">c<sub>8</sub></font>
;<br/>

<a NAME="E5:25_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>3</sub></font>()</span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E6:25_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K3">Union</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>3</sub></font>()
 <b>by </b><i><a class="ref" href="card_3.html#D4">CARD_3:def 4</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">F<sub>3</sub></font>();<br/>



<b>assume </b><a NAME="E7:25_1_2"/>
<font color="Maroon">c<sub>5</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>
 ;<br/>

<a NAME="E8:25_1_2"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> )
 
<b>by </b><i><a class="txt" href="msafree1.html#E3:25_1_2"><i><font color="Green">E18</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E9:25_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">F<sub>3</sub></font>()</span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>4</sub></font>, <a href="card_3.html#K3">Union</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>3</sub></font>()<b> such that </b><br/><a NAME="E10:25_1"/><i><font color="Green">E18</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>(<a class="txt" href="msafree1.html#E8:25_1"><i><font color="Green">E17</font></i></a>);<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K3">Union</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>3</sub></font>() = <font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K3">Union</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>3</sub></font>() = <font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span>;<br/>
<a NAME="E11:25_1"/><i><font color="Green">E19</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span><br/>for <font color="Olive">b<sub>3</sub></font> being   <a href="msafree1.html#M1" target="_self">FinSequence</a> of  <a href="card_3.html#K3">Union</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>3</sub></font>() st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msafree1.html#K1" target="_self">Flatten</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Olive">b<sub>2</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Olive">b<sub>1</sub></font>,<span class="p3">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="msafree1.html#M1" target="_self">FinSequence</a> of  <a href="card_3.html#K3">Union</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>3</sub></font>();<br/>





<a NAME="E1:25_1_3"/><i><font color="Green">E20</font></i>: 
<span class="p1">(<span class="default"><a href="msafree1.html#K1" target="_self">Flatten</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>() <a href="msualg_3.html#K6">#</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="msafree1.html#E1:25_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="msafree1.html#T5" target="_self">Th5</a></i>;<br/>
<b>assume </b><a NAME="E2:25_1_3"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msafree1.html#K1" target="_self">Flatten</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">F<sub>3</sub></font>() <b>by </b><i><a class="txt" href="msafree1.html#E1:25_1_3"><i><font color="Green">E20</font></i></a></i>;<br/>
<a NAME="E4:25_1_3"/><i><font color="Green">E22</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="msafree1.html#E7:25_1"><i><font color="Green">E16</font></i></a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p3">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">F<sub>3</sub></font>()</span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="msafree1.html#E1:25"><i><font color="Green">E9</font></i></a>, <a class="txt" href="msafree1.html#E1:25_1_3"><i><font color="Green">E20</font></i></a>, <a class="txt" href="msafree1.html#E2:25_1_3"><i><font color="Green">E21</font></i></a>, <a class="ref" href="msualg_3.html#D9">MSUALG_3:def 9</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</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>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="msafree1.html#E10:25_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="msafree1.html#E4:25_1_3"><i><font color="Green">E22</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="msafree1.html#E4:25_1_3"><i><font color="Green">E22</font></i></a>, <a class="ref" href="cat_3.html#D1">CAT_3:def 1</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E12:25_1"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree.html#K12">FreeGen</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">F<sub>2</sub></font>() holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:25_1_4"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree.html#K12">FreeGen</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">F<sub>2</sub></font>()
 ;<br/>

<a NAME="E2:25_1_4"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="msafree.html#D18">MSAFREE:def 18</a></i>;<br/>
<a NAME="E3:25_1_4"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E4:25_1_4"/><b>then </b><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K3">Union</a> <span class="p1">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>
 
<b>by </b><i><a class="txt" href="msafree1.html#E2:25_1_4"><i><font color="Green">E22</font></i></a>, <a class="ref" href="card_5.html#T10">CARD_5:10</a></i>;<br/>
<a NAME="E5:25_1_4"/><b>then </b>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>7</sub></font>]
 
<b>by </b><i><a class="txt" href="msafree1.html#E5:25_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="msafree1.html#E1:25_1_4"><i><font color="Green">E21</font></i></a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> = 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="msafree1.html#E2:25"><i><font color="Green">E10</font></i></a>, <a class="txt" href="msafree1.html#E1:25_1_4"><i><font color="Green">E21</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="msafree1.html#E4:25_1_4"><i><font color="Green">E23</font></i></a>, <a class="ref" href="cat_3.html#D1">CAT_3:def 1</a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E13:25_1"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span><br/>for <font color="Olive">b<sub>3</sub></font> being   <a href="msafree1.html#M1" target="_self">FinSequence</a> of  <a href="card_3.html#K3">Union</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>3</sub></font>() st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msafree1.html#K1" target="_self">Flatten</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Olive">b<sub>2</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Olive">b<sub>1</sub></font>,<span class="p3">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="msafree1.html#M1" target="_self">FinSequence</a> of  <a href="card_3.html#K3">Union</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>3</sub></font>();<br/>





<a NAME="E1:25_1_5"/><i><font color="Green">E22</font></i>: 
<span class="p1">(<span class="default"><a href="msafree1.html#K1" target="_self">Flatten</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>() <a href="msualg_3.html#K6">#</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="msafree1.html#E1:25_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="msafree1.html#T5" target="_self">Th5</a></i>;<br/>
<b>assume </b><a NAME="E2:25_1_5"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msafree1.html#K1" target="_self">Flatten</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">F<sub>3</sub></font>() <b>by </b><i><a class="txt" href="msafree1.html#E1:25_1_5"><i><font color="Green">E22</font></i></a></i>;<br/>
<a NAME="E4:25_1_5"/><i><font color="Green">E24</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="msafree1.html#E7:25_1"><i><font color="Green">E16</font></i></a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p3">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">F<sub>3</sub></font>()</span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="msafree1.html#E3:25"><i><font color="Green">E11</font></i></a>, <a class="txt" href="msafree1.html#E1:25_1_5"><i><font color="Green">E22</font></i></a>, <a class="txt" href="msafree1.html#E2:25_1_5"><i><font color="Green">E23</font></i></a>, <a class="ref" href="msualg_3.html#D9">MSUALG_3:def 9</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</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>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="msafree1.html#E10:25_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="msafree1.html#E4:25_1_5"><i><font color="Green">E24</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="msafree1.html#E4:25_1_5"><i><font color="Green">E24</font></i></a>, <a class="ref" href="cat_3.html#D1">CAT_3:def 1</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E14:25_1"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree.html#K12">FreeGen</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">F<sub>2</sub></font>() holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:25_1_6"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree.html#K12">FreeGen</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">F<sub>2</sub></font>()
 ;<br/>

<a NAME="E2:25_1_6"/><b>then </b><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="msafree.html#D18">MSAFREE:def 18</a></i>;<br/>
<a NAME="E3:25_1_6"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E4:25_1_6"/><b>then </b><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K3">Union</a> <span class="p1">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>
 
<b>by </b><i><a class="txt" href="msafree1.html#E2:25_1_6"><i><font color="Green">E24</font></i></a>, <a class="ref" href="card_5.html#T10">CARD_5:10</a></i>;<br/>
<a NAME="E5:25_1_6"/><b>then </b>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>7</sub></font>]
 
<b>by </b><i><a class="txt" href="msafree1.html#E5:25_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="msafree1.html#E1:25_1_6"><i><font color="Green">E23</font></i></a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> = 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="msafree1.html#E4:25"><i><font color="Green">E12</font></i></a>, <a class="txt" href="msafree1.html#E1:25_1_6"><i><font color="Green">E23</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="msafree1.html#E4:25_1_6"><i><font color="Green">E25</font></i></a>, <a class="ref" href="cat_3.html#D1">CAT_3:def 1</a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E15:25_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>from </b><i><a class="ref" href="msafree1.html#S2" target="_self">MSAFREE1:sch 2</a>(<a class="txt" href="msafree1.html#E11:25_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="msafree1.html#E12:25_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="msafree1.html#E13:25_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="msafree1.html#E14:25_1"><i><font color="Green">E22</font></i></a>);<br/></i>
<b>hence </b><a NAME="E16:25_1"/>
<font color="Maroon">F<sub>4</sub></font>() <a href="pboole.html#R6">=</a> <font color="Maroon">F<sub>5</sub></font>()
 ;<br/>


</div>
