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

<a NAME="E1:25_1"/><i><font color="Green" title="E13">A5</font></i>: 
 <a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#G3" title="MSUALG_1:aggr.3">MSAlgebra</a>(# <span class="p1">(<span class="default"><a href="msafree.html#K8" title="MSAFREE:func.8">FreeSort</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>,<span class="p1">(<span class="default"><a href="msafree.html#K10" title="MSAFREE:func.10">FreeOper</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> #)
 
<b>by </b><i><a class="ref" href="msafree.html#D16" title="MSAFREE:def.16">MSAFREE:def 16</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c1">f1</font> = <font color="Maroon">F<sub>4</sub></font>(), <font color="Maroon" title="c2">f2</font> = <font color="Maroon">F<sub>5</sub></font>() as    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of  <a href="msafree.html#K8" title="MSAFREE:func.8">FreeSort</a> <font color="Maroon">F<sub>2</sub></font>(),the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">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" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>for <font color="Olive" title="b1">s</font> being  <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>()  st $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree.html#K12" title="MSAFREE:func.12">FreeGen</a> <font color="Olive" title="b1">s</font>,<font color="Maroon">F<sub>2</sub></font>() holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">s</font>,$2,$1];<br/>
<a NAME="E3:25_1"/><i><font color="Green" title="E14">A6</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="msafree.html#K13" title="MSAFREE:func.13">FreeGen</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon">F<sub>3</sub></font>() &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</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" title="c3">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:25_1_1"/>
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="msafree.html#K13" title="MSAFREE:func.13">FreeGen</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">s</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:25_1_1"/><i><font color="Green" title="E15">A7</font></i>: 
( <font color="Maroon" title="c4">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="msafree.html#K13" title="MSAFREE:func.13">FreeGen</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> &amp; <font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="msafree.html#K13" title="MSAFREE:func.13">FreeGen</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">s</font> )
 <b>by </b><i><a class="ref" href="card_5.html#T10" title="CARD_5:th.10">CARD_5:10</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">s</font> = <font color="Maroon" title="c4">s</font> as   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">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" title="E15">A7</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>take </b><font color="Maroon" title="c6">u</font> = <span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c5">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">e</font>;<br/>

<a NAME="E5:25_1_1"/><i><font color="Green" title="E16">A8</font></i>: 
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree.html#K12" title="MSAFREE:func.12">FreeGen</a> <font color="Maroon" title="c5">s</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" title="E15">A7</font></i></a>, <a class="ref" href="msafree.html#D18" title="MSAFREE:def.18">MSAFREE:def 18</a></i>;<br/>
<a NAME="E6:25_1_1"/>
<font color="Maroon" title="c1">f1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c5">s</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="msafree.html#K8" title="MSAFREE:func.8">FreeSort</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s</font>,the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s</font>
 
;<br/>
<a NAME="E7:25_1_1"/><b>then </b><i><font color="Green" title="E17">A9</font></i>: 
<font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s</font>
 
<b>by </b><i><a class="txt" href="msafree1.html#E5:25_1_1"><i><font color="Green" title="E16">A8</font></i></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/>
<a NAME="E8:25_1_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon">F<sub>3</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>hence </b><a NAME="E9:25_1_1"/>
<font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">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" title="E17">A9</font></i></a>, <a class="ref" href="card_5.html#T10" title="CARD_5:th.10">CARD_5:10</a></i>;<br/>

<b>let </b><font color="Maroon" title="c7">s'</font> be   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">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" title="E18">A10</font></i>: 
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree.html#K12" title="MSAFREE:func.12">FreeGen</a> <font color="Maroon" title="c7">s'</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" title="MSAFREE:func.8">FreeSort</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">s'</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K8" title="MSAFREE:func.8">FreeSort</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="msafree1.html#E5:25_1_1"><i><font color="Green" title="E16">A8</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">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" title="MSAFREE:func.8">FreeSort</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">s'</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a> <span class="p1">(<span class="default"><a href="msafree.html#K8" title="MSAFREE:func.8">FreeSort</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E13:25_1_1"/><b>then </b>
<font color="Maroon" title="c5">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s'</font>
 
<b>by </b><i><a class="ref" href="msafree.html#T13" title="MSAFREE:th.13">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" title="c7">s'</font>,<font color="Maroon" title="c6">u</font>,<font color="Maroon" title="c3">e</font>]
 <b>by </b><i><a class="txt" href="msafree1.html#E2:25"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="msafree1.html#E10:25_1_1"><i><font color="Green" title="E18">A10</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c3">G</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="msafree.html#K13" title="MSAFREE:func.13">FreeGen</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>, <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">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" title="E15">A11</font></i>: 
for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="msafree.html#K13" title="MSAFREE:func.13">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" title="b1">e</font>,<font color="Maroon" title="c3">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1" title="FUNCT_2:sch.1">FUNCT_2:sch 1</a>(<a class="txt" href="msafree1.html#E3:25_1"><i><font color="Green" title="E14">A6</font></i></a>);<br/></i>
<b>consider </b><font color="Maroon" title="c4">R</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:25_1"/><i><font color="Green" title="E16">A12</font></i>: 
<font color="Maroon" title="c4">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">o</font>,<font color="Olive" title="b2">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">o</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Olive" title="b2">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Olive" title="b1">o</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" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>for <font color="Olive" title="b1">o</font> being  <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive" title="b2">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Olive" title="b1">o</font>,<font color="Maroon">F<sub>3</sub></font>()  st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">o</font>,<font color="Olive" title="b2">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Olive" title="b1">o</font>,<font color="Maroon">F<sub>3</sub></font>()</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">a</font>;<br/>
<a NAME="E8:25_1"/><i><font color="Green" title="E17">A13</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R</font> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon">F<sub>3</sub></font>() &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</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" title="c5">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:25_1_2"/>
<font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">o</font> being   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon" title="c7">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c6">o</font>,<font color="Maroon">F<sub>3</sub></font>()<b> such that </b><br/><a NAME="E3:25_1_2"/><i><font color="Green" title="E18">A14</font></i>: 
<font color="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c7">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="msafree1.html#E7:25_1"><i><font color="Green" title="E16">A12</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">u</font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c6">o</font>,<font color="Maroon">F<sub>3</sub></font>()</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<b>take </b>
<font color="Maroon" title="c8">u</font>
;<br/>

<a NAME="E5:25_1_2"/>
<font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon">F<sub>3</sub></font>()</span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E6:25_1_2"/>
<font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon">F<sub>3</sub></font>()
 <b>by </b><i><a class="ref" href="card_3.html#D4" title="CARD_3:def.4">CARD_3:def 4</a></i>;<br/>

<b>let </b><font color="Maroon" title="c9">o'</font> be   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>let </b><font color="Maroon" title="c10">x'</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c9">o'</font>,<font color="Maroon">F<sub>3</sub></font>();<br/>



<b>assume </b><a NAME="E7:25_1_2"/>
<font color="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">o'</font>,<font color="Maroon" title="c10">x'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ;<br/>

<a NAME="E8:25_1_2"/><b>then </b>
( <font color="Maroon" title="c6">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">o'</font> &amp; <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x'</font> )
 
<b>by </b><i><a class="txt" href="msafree1.html#E3:25_1_2"><i><font color="Green" title="E18">A14</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E9:25_1_2"/>
<font color="Maroon" title="c8">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c9">o'</font>,<font color="Maroon">F<sub>3</sub></font>()</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x'</font>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c5">H</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c4">R</font>, <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">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" title="E18">A15</font></i>: 
for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">e</font>,<font color="Maroon" title="c5">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1" title="FUNCT_2:sch.1">FUNCT_2:sch 1</a>(<a class="txt" href="msafree1.html#E8:25_1"><i><font color="Green" title="E17">A13</font></i></a>);<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon">F<sub>3</sub></font>() = <font color="Maroon" title="c3">G</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> $1;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon">F<sub>3</sub></font>() = <font color="Maroon" title="c5">H</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">$1,$3</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<a NAME="E11:25_1"/><i><font color="Green" title="E19">A16</font></i>: 
for <font color="Olive" title="b1">o</font> being  <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive" title="b2">ts</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Olive" title="b1">o</font>,<span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span><br/> for <font color="Olive" title="b3">x</font> being   <a href="msafree1.html#M1" title="MSAFREE1:mode.1">FinSequence</a> of  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon">F<sub>3</sub></font>()  st <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msafree1.html#K1" title="MSAFREE1:func.1">Flatten</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b2">ts</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Olive" title="b1">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Olive" title="b1">o</font>,<span class="p3">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">ts</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">o</font>,<font color="Olive" title="b2">ts</font>,<font color="Olive" title="b3">x</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" title="c6">o</font> be   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>let </b><font color="Maroon" title="c7">ts</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c6">o</font>,<span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>;<br/><b>let </b><font color="Maroon" title="c8">x</font> be    <a href="msafree1.html#M1" title="MSAFREE1:mode.1">FinSequence</a> of  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon">F<sub>3</sub></font>();<br/>





<a NAME="E1:25_1_3"/><i><font color="Green" title="E20">A17</font></i>: 
<span class="p1">(<span class="default"><a href="msafree1.html#K1" title="MSAFREE1:func.1">Flatten</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">ts</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>() <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c7">ts</font>
 
<b>by </b><i><a class="txt" href="msafree1.html#E1:25_1"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="msafree1.html#T5" target="_self" title="MSAFREE1:th.5">Th5</a></i>;<br/>
<b>assume </b><a NAME="E2:25_1_3"/><i><font color="Green" title="E21">A18</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msafree1.html#K1" title="MSAFREE1:func.1">Flatten</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">ts</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c9">a</font> = <font color="Maroon" title="c8">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c6">o</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" title="E20">A17</font></i></a></i>;<br/>
<a NAME="E4:25_1_3"/><i><font color="Green" title="E22">A19</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c9">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R</font>
 
<b>by </b><i><a class="txt" href="msafree1.html#E7:25_1"><i><font color="Green" title="E16">A12</font></i></a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c6">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c6">o</font>,<span class="p3">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">ts</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c6">o</font>,<font color="Maroon">F<sub>3</sub></font>()</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">a</font>
<b>by </b><i><a class="txt" href="msafree1.html#E1:25"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="msafree1.html#E1:25_1_3"><i><font color="Green" title="E20">A17</font></i></a>, <a class="txt" href="msafree1.html#E2:25_1_3"><i><font color="Green" title="E21">A18</font></i></a>, <a class="ref" href="msualg_3.html#D9" title="MSUALG_3:def.9">MSUALG_3:def 9</a></i>
<br/>.= 
<font color="Maroon" title="c5">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c8">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="msafree1.html#E10:25_1"><i><font color="Green" title="E18">A15</font></i></a>, <a class="txt" href="msafree1.html#E4:25_1_3"><i><font color="Green" title="E22">A19</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">H</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c8">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="msafree1.html#E4:25_1_3"><i><font color="Green" title="E22">A19</font></i></a>, <a class="ref" href="cat_3.html#D1" title="CAT_3:def.1">CAT_3:def 1</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E12:25_1"/><i><font color="Green" title="E20">A20</font></i>: 
for <font color="Olive" title="b1">s</font> being  <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree.html#K12" title="MSAFREE:func.12">FreeGen</a> <font color="Olive" title="b1">s</font>,<font color="Maroon">F<sub>2</sub></font>() holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Olive" title="b1">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">y</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" title="c6">s</font> be   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>let </b><font color="Maroon" title="c7">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><a NAME="E1:25_1_4"/><i><font color="Green" title="E21">A21</font></i>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree.html#K12" title="MSAFREE:func.12">FreeGen</a> <font color="Maroon" title="c6">s</font>,<font color="Maroon">F<sub>2</sub></font>()
 ;<br/>

<a NAME="E2:25_1_4"/><b>then </b><i><font color="Green" title="E22">A22</font></i>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="msafree.html#K13" title="MSAFREE:func.13">FreeGen</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s</font>
 
<b>by </b><i><a class="ref" href="msafree.html#D18" title="MSAFREE:def.18">MSAFREE:def 18</a></i>;<br/>
<a NAME="E3:25_1_4"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="msafree.html#K13" title="MSAFREE:func.13">FreeGen</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E4:25_1_4"/><b>then </b><i><font color="Green" title="E23">A23</font></i>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="msafree.html#K13" title="MSAFREE:func.13">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" title="E22">A22</font></i></a>, <a class="ref" href="card_5.html#T10" title="CARD_5:th.10">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" title="c6">s</font>,<font color="Maroon" title="c3">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c7">y</font>]
 
<b>by </b><i><a class="txt" href="msafree1.html#E5:25_1"><i><font color="Green" title="E15">A11</font></i></a>, <a class="txt" href="msafree1.html#E1:25_1_4"><i><font color="Green" title="E21">A21</font></i></a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c6">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">y</font> = 
<font color="Maroon" title="c3">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">y</font>
<b>by </b><i><a class="txt" href="msafree1.html#E2:25"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="msafree1.html#E1:25_1_4"><i><font color="Green" title="E21">A21</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">G</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c7">y</font>
<b>by </b><i><a class="txt" href="msafree1.html#E4:25_1_4"><i><font color="Green" title="E23">A23</font></i></a>, <a class="ref" href="cat_3.html#D1" title="CAT_3:def.1">CAT_3:def 1</a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E13:25_1"/><i><font color="Green" title="E21">A24</font></i>: 
for <font color="Olive" title="b1">o</font> being  <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive" title="b2">ts</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Olive" title="b1">o</font>,<span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span><br/> for <font color="Olive" title="b3">x</font> being   <a href="msafree1.html#M1" title="MSAFREE1:mode.1">FinSequence</a> of  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon">F<sub>3</sub></font>()  st <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msafree1.html#K1" title="MSAFREE1:func.1">Flatten</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b2">ts</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Olive" title="b1">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Olive" title="b1">o</font>,<span class="p3">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">ts</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">o</font>,<font color="Olive" title="b2">ts</font>,<font color="Olive" title="b3">x</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" title="c6">o</font> be   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>let </b><font color="Maroon" title="c7">ts</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c6">o</font>,<span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>;<br/><b>let </b><font color="Maroon" title="c8">x</font> be    <a href="msafree1.html#M1" title="MSAFREE1:mode.1">FinSequence</a> of  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon">F<sub>3</sub></font>();<br/>





<a NAME="E1:25_1_5"/><i><font color="Green" title="E22">A25</font></i>: 
<span class="p1">(<span class="default"><a href="msafree1.html#K1" title="MSAFREE1:func.1">Flatten</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">ts</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>() <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c7">ts</font>
 
<b>by </b><i><a class="txt" href="msafree1.html#E1:25_1"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="msafree1.html#T5" target="_self" title="MSAFREE1:th.5">Th5</a></i>;<br/>
<b>assume </b><a NAME="E2:25_1_5"/><i><font color="Green" title="E23">A26</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msafree1.html#K1" title="MSAFREE1:func.1">Flatten</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">ts</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c9">a</font> = <font color="Maroon" title="c8">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c6">o</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" title="E22">A25</font></i></a></i>;<br/>
<a NAME="E4:25_1_5"/><i><font color="Green" title="E24">A27</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c9">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R</font>
 
<b>by </b><i><a class="txt" href="msafree1.html#E7:25_1"><i><font color="Green" title="E16">A12</font></i></a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c6">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c6">o</font>,<span class="p3">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">ts</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c6">o</font>,<font color="Maroon">F<sub>3</sub></font>()</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">a</font>
<b>by </b><i><a class="txt" href="msafree1.html#E3:25"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="msafree1.html#E1:25_1_5"><i><font color="Green" title="E22">A25</font></i></a>, <a class="txt" href="msafree1.html#E2:25_1_5"><i><font color="Green" title="E23">A26</font></i></a>, <a class="ref" href="msualg_3.html#D9" title="MSUALG_3:def.9">MSUALG_3:def 9</a></i>
<br/>.= 
<font color="Maroon" title="c5">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c8">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="msafree1.html#E10:25_1"><i><font color="Green" title="E18">A15</font></i></a>, <a class="txt" href="msafree1.html#E4:25_1_5"><i><font color="Green" title="E24">A27</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">H</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c8">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="msafree1.html#E4:25_1_5"><i><font color="Green" title="E24">A27</font></i></a>, <a class="ref" href="cat_3.html#D1" title="CAT_3:def.1">CAT_3:def 1</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E14:25_1"/><i><font color="Green" title="E22">A28</font></i>: 
for <font color="Olive" title="b1">s</font> being  <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree.html#K12" title="MSAFREE:func.12">FreeGen</a> <font color="Olive" title="b1">s</font>,<font color="Maroon">F<sub>2</sub></font>() holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Olive" title="b1">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">y</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" title="c6">s</font> be   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>let </b><font color="Maroon" title="c7">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><a NAME="E1:25_1_6"/><i><font color="Green" title="E23">A29</font></i>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree.html#K12" title="MSAFREE:func.12">FreeGen</a> <font color="Maroon" title="c6">s</font>,<font color="Maroon">F<sub>2</sub></font>()
 ;<br/>

<a NAME="E2:25_1_6"/><b>then </b><i><font color="Green" title="E24">A30</font></i>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="msafree.html#K13" title="MSAFREE:func.13">FreeGen</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s</font>
 
<b>by </b><i><a class="ref" href="msafree.html#D18" title="MSAFREE:def.18">MSAFREE:def 18</a></i>;<br/>
<a NAME="E3:25_1_6"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="msafree.html#K13" title="MSAFREE:func.13">FreeGen</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E4:25_1_6"/><b>then </b><i><font color="Green" title="E25">A31</font></i>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="msafree.html#K13" title="MSAFREE:func.13">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" title="E24">A30</font></i></a>, <a class="ref" href="card_5.html#T10" title="CARD_5:th.10">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" title="c6">s</font>,<font color="Maroon" title="c3">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c7">y</font>]
 
<b>by </b><i><a class="txt" href="msafree1.html#E5:25_1"><i><font color="Green" title="E15">A11</font></i></a>, <a class="txt" href="msafree1.html#E1:25_1_6"><i><font color="Green" title="E23">A29</font></i></a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c6">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">y</font> = 
<font color="Maroon" title="c3">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">y</font>
<b>by </b><i><a class="txt" href="msafree1.html#E4:25"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="msafree1.html#E1:25_1_6"><i><font color="Green" title="E23">A29</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">G</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c7">y</font>
<b>by </b><i><a class="txt" href="msafree1.html#E4:25_1_6"><i><font color="Green" title="E25">A31</font></i></a>, <a class="ref" href="cat_3.html#D1" title="CAT_3:def.1">CAT_3:def 1</a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E15:25_1"/>
<font color="Maroon" title="c1">f1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c2">f2</font>
 
<b>from </b><i><a class="ref" href="msafree1.html#S2" target="_self" title="MSAFREE1:sch.2">MSAFREE1:sch 2</a>(<a class="txt" href="msafree1.html#E11:25_1"><i><font color="Green" title="E19">A16</font></i></a>, <a class="txt" href="msafree1.html#E12:25_1"><i><font color="Green" title="E20">A20</font></i></a>, <a class="txt" href="msafree1.html#E13:25_1"><i><font color="Green" title="E21">A24</font></i></a>, <a class="txt" href="msafree1.html#E14:25_1"><i><font color="Green" title="E22">A28</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" title="PBOOLE:pred.6">=</a> <font color="Maroon">F<sub>5</sub></font>()
 ;<br/>


</div>
