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

<b>let </b><font color="Maroon" title="c1">S</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> non <a href="msualg_1.html#V2" title="MSUALG_1:attr.2">void</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon" title="c2">X</font> be  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V2</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>;<br/><b>let </b><font color="Maroon" title="c3">v</font> be   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S</font>;<br/>





<b>set </b><font color="Maroon" title="c4">SF</font> = the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c5">d0</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">t</font> where <font color="Olive" title="b1">t</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> :  <a href="msafree2.html#K9" title="MSAFREE2:func.9">depth</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 </span>}</span> ;<br/>
<a NAME="E1:50"/><i><font color="Green" title="E24">A1</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">t</font> where <font color="Olive" title="b1">t</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> :  <a href="msafree2.html#K9" title="MSAFREE2:func.9">depth</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeGen</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="msualg_2.html#K1" title="MSUALG_2:func.1">Constants</a> <span class="p2">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c3">v</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:50_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">t</font> where <font color="Olive" title="b1">t</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> :  <a href="msafree2.html#K9" title="MSAFREE2:func.9">depth</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">t</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font><b> such that </b><br/><a NAME="E3:50_1"/><i><font color="Green" title="E25">A2</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t</font> &amp;  <a href="msafree2.html#K9" title="MSAFREE2:func.9">depth</a> <font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 ;<br/>
<b>consider </b><font color="Maroon" title="c8">dt</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="trees_2.html#NM4" title="TREES_2:NM.4">DecoratedTree</a>, <font color="Maroon" title="c9">ft</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a><b> such that </b><br/><a NAME="E5:50_1"/><i><font color="Green" title="E26">A3</font></i>: 
( <font color="Maroon" title="c8">dt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t</font> &amp; <font color="Maroon" title="c9">ft</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c8">dt</font> &amp;  <a href="msafree2.html#K9" title="MSAFREE2:func.9">depth</a> <font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <font color="Maroon" title="c9">ft</font> )
 <b>by </b><i><a class="ref" href="msafree2.html#D14" title="MSAFREE2:def.14">MSAFREE2:def 14</a></i>;<br/>
<a NAME="E6:50_1"/>
<font color="Maroon" title="c7">t</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 <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font>
 
;<br/>
<a NAME="E7:50_1"/><b>then </b>
<font color="Maroon" title="c7">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree.html#K7" title="MSAFREE:func.7">FreeSort</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">v</font>
 
<b>by </b><i><a class="ref" href="msafree.html#D13" title="MSAFREE:def.13">MSAFREE:def 13</a></i>;<br/>
<a NAME="E8:50_1"/><b>then </b>
<font color="Maroon" title="c7">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="trees_9.html#M1" title="TREES_9:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K5" title="MSAFREE:func.5">DTConMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> : (  ex <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">x</font>,<font color="Maroon" title="c3">v</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) or  ex <font color="Olive" title="b2">o</font> being  <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Olive" title="b2">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> ) ) </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon" title="c10">a</font> being    <a href="trees_9.html#M1" title="TREES_9:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K5" title="MSAFREE:func.5">DTConMSA</a> <font color="Maroon" title="c2">X</font></span>)</span><b> such that </b><br/><a NAME="E10:50_1"/><i><font color="Green" title="E27">A4</font></i>: 
( <font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a</font> &amp; (  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> &amp; <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">v</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) or  ex <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" title="c1">S</font> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Olive" title="b1">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> ) ) )
 ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:50_1_1"/>
(  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> &amp; <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">v</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) or  ex <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" title="c1">S</font> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Olive" title="b1">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> ) )
 </span><b>by </b><i><a class="txt" href="msscyc_1.html#E10:50_1"><i><font color="Green" title="E27">A4</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_1"><b>suppose </b></a><a NAME="E1:50_1_1_1"/>
 ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> &amp; <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">v</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c11">x1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:50_1_1_1"/><i><font color="Green" title="E28">A5</font></i>: 
( <font color="Maroon" title="c11">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> &amp; <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x1</font>,<font color="Maroon" title="c3">v</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 ;<br/><a NAME="E4:50_1_1_1"/>
<font color="Maroon" title="c7">t</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="c3">v</font>,<font color="Maroon" title="c2">X</font>
 <b>by </b><i><a class="txt" href="msscyc_1.html#E10:50_1"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="msscyc_1.html#E3:50_1_1_1"><i><font color="Green" title="E28">A5</font></i></a>, <a class="ref" href="msafree.html#D17" title="MSAFREE:def.17">MSAFREE:def 17</a></i>;<br/><b>hence </b><a NAME="E5:50_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeGen</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="msualg_2.html#K1" title="MSUALG_2:func.1">Constants</a> <span class="p2">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c3">v</font></span>)</span>
 <b>by </b><i><a class="txt" href="msscyc_1.html#E3:50_1"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_2"><b>suppose </b></a><a NAME="E1:50_1_1_2"/>
 ex <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" title="c1">S</font> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Olive" title="b1">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c11">o</font> being   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E3:50_1_1_2"/><i><font color="Green" title="E28">A6</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c11">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> )
 ;<br/><b>reconsider </b><font color="Maroon" title="c12">t'</font> = <font color="Maroon" title="c7">t</font> as   <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">X</font> <b>by </b><i><a class="txt" href="msscyc_1.html#E10:50_1"><i><font color="Green" title="E27">A4</font></i></a>, <a class="ref" href="msaterm.html#D1" title="MSATERM:def.1">MSATERM:def 1</a></i>;<br/><b>consider </b><font color="Maroon" title="c13">ars</font> being    <a href="msaterm.html#M1" title="MSATERM:mode.1">ArgumentSeq</a> of  <a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c2">X</font><b> such that </b><br/><a NAME="E6:50_1_1_2"/><i><font color="Green" title="E29">A7</font></i>: 
<font color="Maroon" title="c12">t'</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="c11">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c13">ars</font>
 <b>by </b><i><a class="txt" href="msscyc_1.html#E10:50_1"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="msscyc_1.html#E3:50_1_1_2"><i><font color="Green" title="E28">A6</font></i></a>, <a class="ref" href="msaterm.html#T10" title="MSATERM:th.10">MSATERM:10</a></i>;<br/><a NAME="E7:50_1_1_2"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c8">dt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 0
 <b>by </b><i><a class="txt" href="msscyc_1.html#E3:50_1"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="msscyc_1.html#E5:50_1"><i><font color="Green" title="E26">A3</font></i></a>, <a class="ref" href="trees_1.html#T80" title="TREES_1:th.80">TREES_1:80</a></i>;<br/><a NAME="E8:50_1_1_2"/><b>then </b>
<font color="Maroon" title="c8">dt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">dt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span>
 <b>by </b><i><a class="ref" href="trees_4.html#T5" title="TREES_4:th.5">TREES_4:5</a></i>;<br/><a NAME="E9:50_1_1_2"/><b>then </b><i><font color="Green" title="E30">A8</font></i>: 
<font color="Maroon" title="c13">ars</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="msscyc_1.html#E5:50_1"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="msscyc_1.html#E6:50_1_1_2"><i><font color="Green" title="E29">A7</font></i></a>, <a class="ref" href="trees_4.html#T17" title="TREES_4:th.17">TREES_4:17</a></i>;<br/><b>then </b>0 = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">ars</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c11">o</font></span>)</span>
<b>by </b><i><a class="ref" href="msaterm.html#T22" title="MSATERM:th.22">MSATERM:22</a></i>
;<br/><a NAME="E11:50_1_1_2"/><b>then </b>
 <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c11">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/><a NAME="E12:50_1_1_2"/><b>then </b><i><font color="Green" title="E31">A9</font></i>: 
the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c1">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="msualg_1.html#D6" title="MSUALG_1:def.6">MSUALG_1:def 6</a></i>;<br/><a NAME="E13:50_1_1_2"/><i><font color="Green" title="E32">A10</font></i>: 
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c1">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font>
 <b>by </b><i><a class="txt" href="msscyc_1.html#E3:50_1_1_2"><i><font color="Green" title="E28">A6</font></i></a>, <a class="ref" href="msualg_1.html#D7" title="MSUALG_1:def.7">MSUALG_1:def 7</a></i>;<br/><i><font color="Green" title="E33">A11</font></i>:  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c11">o</font>,<span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="msafree.html#K10" title="MSAFREE:func.10">FreeOper</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c11">o</font>
<b>by </b><i><a class="ref" href="msualg_1.html#D11" title="MSUALG_1:def.11">MSUALG_1:def 11</a></i>
<br/>.= 
 <a href="msafree.html#K9" title="MSAFREE:func.9">DenOp</a> <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c2">X</font>
<b>by </b><i><a class="ref" href="msafree.html#D15" title="MSAFREE:def.15">MSAFREE:def 15</a></i>
;<br/><a NAME="E15:50_1_1_2"/><i><font color="Green" title="E34">A12</font></i>: 
 <a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c13">ars</font>
 <b>by </b><i><a class="ref" href="msaterm.html#T21" title="MSATERM:th.21">MSATERM:21</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c14">ars'</font> = <font color="Maroon" title="c13">ars</font> as    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K5" title="MSAFREE:func.5">DTConMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <b>by </b><i><a class="txt" href="msscyc_1.html#E9:50_1_1_2"><i><font color="Green" title="E30">A8</font></i></a>, <a class="ref" href="finseq_1.html#T29" title="FINSEQ_1:th.29">FINSEQ_1:29</a></i>;<br/><a NAME="E17:50_1_1_2"/><i><font color="Green" title="E35">A13</font></i>: 
<span class="p1">(<span class="default"><a href="msafree.html#K9" title="MSAFREE:func.9">DenOp</a> <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">ars'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t</font>
 <b>by </b><i><a class="txt" href="msscyc_1.html#E15:50_1_1_2"><i><font color="Green" title="E34">A12</font></i></a>, <a class="ref" href="msafree.html#D14" title="MSAFREE:def.14">MSAFREE:def 14</a>, <a class="txt" href="msscyc_1.html#E6:50_1_1_2"><i><font color="Green" title="E29">A7</font></i></a></i>;<br/><a NAME="E18:50_1_1_2"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c11">o</font>,<span class="p2">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="msscyc_1.html#E12:50_1_1_2"><i><font color="Green" title="E31">A9</font></i></a>, <a class="ref" href="msscyc_1.html#T25" target="_self" title="MSSCYC_1:th.25">Th25</a></i>;<br/><a NAME="E19:50_1_1_2"/><b>then </b>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <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="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c11">o</font>,<span class="p2">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E20:50_1_1_2"/><b>then </b><i><font color="Green" title="E36">A14</font></i>: 
<font color="Maroon" title="c7">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c11">o</font>,<span class="p2">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="msscyc_1.html#E9:50_1_1_2"><i><font color="Green" title="E30">A8</font></i></a>, <a class="txt" href="msscyc_1.html#E14:50_1_1_2"><i><font color="Green" title="E33">A11</font></i></a>, <a class="txt" href="msscyc_1.html#E17:50_1_1_2"><i><font color="Green" title="E35">A13</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon" title="c15">Av</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E22:50_1_1_2"/><i><font color="Green" title="E37">A15</font></i>: 
( <font color="Maroon" title="c15">Av</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> &amp;  <a href="msualg_2.html#K1" title="MSUALG_2:func.1">Constants</a> <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a1</font> where <font color="Olive" title="b1">a1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c15">Av</font> :  ex <font color="Olive" title="b2">o</font> being  <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font> st <br/>( the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c1">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c1">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> &amp; <font color="Olive" title="b1">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Olive" title="b2">o</font>,<span class="p2">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="msualg_2.html#D4" title="MSUALG_2:def.4">MSUALG_2:def 4</a></i>;<br/><a NAME="E23:50_1_1_2"/>
<font color="Maroon" title="c7">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_2.html#K1" title="MSUALG_2:func.1">Constants</a> <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c3">v</font>
 <b>by </b><i><a class="txt" href="msscyc_1.html#E12:50_1_1_2"><i><font color="Green" title="E31">A9</font></i></a>, <a class="txt" href="msscyc_1.html#E13:50_1_1_2"><i><font color="Green" title="E32">A10</font></i></a>, <a class="txt" href="msscyc_1.html#E20:50_1_1_2"><i><font color="Green" title="E36">A14</font></i></a>, <a class="txt" href="msscyc_1.html#E22:50_1_1_2"><i><font color="Green" title="E37">A15</font></i></a></i>;<br/><b>hence </b><a NAME="E24:50_1_1_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeGen</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="msualg_2.html#K1" title="MSUALG_2:func.1">Constants</a> <span class="p2">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c3">v</font></span>)</span>
 <b>by </b><i><a class="txt" href="msscyc_1.html#E3:50_1"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E2:50"/><i><font color="Green" title="E25">A16</font></i>: 
 <a href="msafree.html#K12" title="MSAFREE:func.12">FreeGen</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">t</font> where <font color="Olive" title="b1">t</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> :  <a href="msafree2.html#K9" title="MSAFREE2:func.9">depth</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:50_2"/><i><font color="Green" title="E26">A17</font></i>: 
<font color="Maroon" title="c6">x</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="c3">v</font>,<font color="Maroon" title="c2">X</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:50_2"/><i><font color="Green" title="E27">A18</font></i>: 
( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c3">v</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="msafree.html#D17" title="MSAFREE:def.17">MSAFREE:def 17</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">x'</font> = <font color="Maroon" title="c6">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> <b>by </b><i><a class="txt" href="msscyc_1.html#E1:50_2"><i><font color="Green" title="E26">A17</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">dt</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="trees_2.html#NM4" title="TREES_2:NM.4">DecoratedTree</a>, <font color="Maroon" title="c10">t</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a><b> such that </b><br/><a NAME="E6:50_2"/><i><font color="Green" title="E28">A19</font></i>: 
( <font color="Maroon" title="c9">dt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x'</font> &amp; <font color="Maroon" title="c10">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">dt</font> &amp;  <a href="msafree2.html#K9" title="MSAFREE2:func.9">depth</a> <font color="Maroon" title="c8">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <font color="Maroon" title="c10">t</font> )
 <b>by </b><i><a class="ref" href="msafree2.html#D14" title="MSAFREE2:def.14">MSAFREE2:def 14</a></i>;<br/>
<a NAME="E7:50_2"/>
 <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <font color="Maroon" title="c10">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="txt" href="msscyc_1.html#E3:50_2"><i><font color="Green" title="E27">A18</font></i></a>, <a class="txt" href="msscyc_1.html#E6:50_2"><i><font color="Green" title="E28">A19</font></i></a>, <a class="ref" href="trees_1.html#T79" title="TREES_1:th.79">TREES_1:79</a>, <a class="ref" href="trees_4.html#T3" title="TREES_4:th.3">TREES_4:3</a></i>;<br/>
<b>hence </b><a NAME="E8:50_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">t</font> where <font color="Olive" title="b1">t</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> :  <a href="msafree2.html#K9" title="MSAFREE2:func.9">depth</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 </span>}</span> 
 <b>by </b><i><a class="txt" href="msscyc_1.html#E6:50_2"><i><font color="Green" title="E28">A19</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:50"/>
 <a href="msualg_2.html#K1" title="MSUALG_2:func.1">Constants</a> <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c3">v</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">t</font> where <font color="Olive" title="b1">t</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> :  <a href="msafree2.html#K9" title="MSAFREE2:func.9">depth</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:50_3"/><i><font color="Green" title="E26">A20</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_2.html#K1" title="MSUALG_2:func.1">Constants</a> <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c3">v</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c7">Av</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:50_3"/><i><font color="Green" title="E27">A21</font></i>: 
( <font color="Maroon" title="c7">Av</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> &amp;  <a href="msualg_2.html#K1" title="MSUALG_2:func.1">Constants</a> <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">Av</font> :  ex <font color="Olive" title="b2">o</font> being  <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font> st <br/>( the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c1">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c1">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Olive" title="b2">o</font>,<span class="p2">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="msualg_2.html#D4" title="MSUALG_2:def.4">MSUALG_2:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">Av</font><b> such that </b><br/><a NAME="E5:50_3"/><i><font color="Green" title="E28">A22</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font> &amp;  ex <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" title="c1">S</font> st <br/>( the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c1">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c1">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> &amp; <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <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>,<span class="p2">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="msscyc_1.html#E1:50_3"><i><font color="Green" title="E26">A20</font></i></a>, <a class="txt" href="msscyc_1.html#E3:50_3"><i><font color="Green" title="E27">A21</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">o</font> being   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E7:50_3"/><i><font color="Green" title="E29">A23</font></i>: 
( the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c1">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c1">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> &amp; <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <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>,<span class="p2">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="msscyc_1.html#E5:50_3"><i><font color="Green" title="E28">A22</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">a</font> = <font color="Maroon" title="c8">a</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> <b>by </b><i><a class="txt" href="msscyc_1.html#E3:50_3"><i><font color="Green" title="E27">A21</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">dt</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="trees_2.html#NM4" title="TREES_2:NM.4">DecoratedTree</a>, <font color="Maroon" title="c12">t</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a><b> such that </b><br/><a NAME="E10:50_3"/><i><font color="Green" title="E30">A24</font></i>: 
( <font color="Maroon" title="c11">dt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a</font> &amp; <font color="Maroon" title="c12">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c11">dt</font> &amp;  <a href="msafree2.html#K9" title="MSAFREE2:func.9">depth</a> <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <font color="Maroon" title="c12">t</font> )
 <b>by </b><i><a class="ref" href="msafree2.html#D14" title="MSAFREE2:def.14">MSAFREE2:def 14</a></i>;<br/>
<a NAME="E11:50_3"/><i><font color="Green" title="E31">A25</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</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>,<span class="p2">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="msscyc_1.html#E7:50_3"><i><font color="Green" title="E29">A23</font></i></a>, <a class="ref" href="msscyc_1.html#T25" target="_self" title="MSSCYC_1:th.25">Th25</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">d</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E13:50_3"/><i><font color="Green" title="E32">A26</font></i>: 
( <font color="Maroon" title="c13">d</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="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c9">o</font>,<span class="p2">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c10">a</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>,<span class="p2">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">d</font> )
 <b>by </b><i><a class="txt" href="msscyc_1.html#E7:50_3"><i><font color="Green" title="E29">A23</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E14:50_3"/><i><font color="Green" title="E33">A27</font></i>: 
<font color="Maroon" title="c13">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="msscyc_1.html#E11:50_3"><i><font color="Green" title="E31">A25</font></i></a>, <a class="txt" href="msscyc_1.html#E13:50_3"><i><font color="Green" title="E32">A26</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<i><font color="Green" title="E34">A28</font></i>:  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c9">o</font>,<span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="msafree.html#K10" title="MSAFREE:func.10">FreeOper</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c9">o</font>
<b>by </b><i><a class="ref" href="msualg_1.html#D11" title="MSUALG_1:def.11">MSUALG_1:def 11</a></i>
<br/>.= 
 <a href="msafree.html#K9" title="MSAFREE:func.9">DenOp</a> <font color="Maroon" title="c9">o</font>,<font color="Maroon" title="c2">X</font>
<b>by </b><i><a class="ref" href="msafree.html#D15" title="MSAFREE:def.15">MSAFREE:def 15</a></i>
;<br/>
<b>reconsider </b><font color="Maroon" title="c14">p</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K5" title="MSAFREE:func.5">DTConMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <b>by </b><i><a class="ref" href="finseq_1.html#T29" title="FINSEQ_1:th.29">FINSEQ_1:29</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msafree.html#K8" title="MSAFREE:func.8">FreeSort</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">o</font> = 
 <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c9">o</font>,<span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span>
<b>by </b><i><a class="ref" href="msualg_1.html#D9" title="MSUALG_1:def.9">MSUALG_1:def 9</a></i>
<br/>.= 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</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>,<span class="p2">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
;<br/>
<a NAME="E18:50_3"/><b>then </b>
<font color="Maroon" title="c14">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msafree.html#K8" title="MSAFREE:func.8">FreeSort</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">o</font>
 
<b>by </b><i><a class="txt" href="msscyc_1.html#E11:50_3"><i><font color="Green" title="E31">A25</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E19:50_3"/><b>then </b>
 <a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> <font color="Maroon" title="c9">o</font>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c14">p</font>
 
<b>by </b><i><a class="ref" href="msafree.html#T10" title="MSAFREE:th.10">MSAFREE:10</a></i>;<br/>
<a NAME="E20:50_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="msafree.html#K9" title="MSAFREE:func.9">DenOp</a> <font color="Maroon" title="c9">o</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> <font color="Maroon" title="c9">o</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c14">p</font>
 
<b>by </b><i><a class="ref" href="msafree.html#D14" title="MSAFREE:def.14">MSAFREE:def 14</a></i>;<br/>
<a NAME="E21:50_3"/><b>then </b>
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <span class="p1">(<span class="default"><a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> <font color="Maroon" title="c9">o</font>,<font color="Maroon" title="c2">X</font></span>)</span>
 
<b>by </b><i><a class="txt" href="msscyc_1.html#E13:50_3"><i><font color="Green" title="E32">A26</font></i></a>, <a class="txt" href="msscyc_1.html#E14:50_3"><i><font color="Green" title="E33">A27</font></i></a>, <a class="txt" href="msscyc_1.html#E15:50_3"><i><font color="Green" title="E34">A28</font></i></a>, <a class="ref" href="trees_4.html#T20" title="TREES_4:th.20">TREES_4:20</a></i>;<br/>
<a NAME="E22:50_3"/><b>then </b>
 <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <font color="Maroon" title="c12">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="txt" href="msscyc_1.html#E10:50_3"><i><font color="Green" title="E30">A24</font></i></a>, <a class="ref" href="trees_1.html#T79" title="TREES_1:th.79">TREES_1:79</a>, <a class="ref" href="trees_4.html#T3" title="TREES_4:th.3">TREES_4:3</a></i>;<br/>
<b>hence </b><a NAME="E23:50_3"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">t</font> where <font color="Olive" title="b1">t</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> :  <a href="msafree2.html#K9" title="MSAFREE2:func.9">depth</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 </span>}</span> 
 <b>by </b><i><a class="txt" href="msscyc_1.html#E5:50_3"><i><font color="Green" title="E28">A22</font></i></a>, <a class="txt" href="msscyc_1.html#E10:50_3"><i><font color="Green" title="E30">A24</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:50"/><b>then </b>
<span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeGen</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="msualg_2.html#K1" title="MSUALG_2:func.1">Constants</a> <span class="p2">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c3">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">t</font> where <font color="Olive" title="b1">t</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> :  <a href="msafree2.html#K9" title="MSAFREE2:func.9">depth</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 </span>}</span> 
 
<b>by </b><i><a class="txt" href="msscyc_1.html#E2:50"><i><font color="Green" title="E25">A16</font></i></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/>
<b>hence </b><a NAME="E5:50"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">t</font> where <font color="Olive" title="b1">t</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> :  <a href="msafree2.html#K9" title="MSAFREE2:func.9">depth</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeGen</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="msualg_2.html#K1" title="MSUALG_2:func.1">Constants</a> <span class="p2">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c3">v</font></span>)</span>
 <b>by </b><i><a class="txt" href="msscyc_1.html#E1:50"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div>
