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

<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon">X</font> be  <a href="relat_1.html#V2">V2</a> <a href="finset_1.html#V2">V24</a>  <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>;<br/>



<b>assume </b><a NAME="E1:8"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">S</font> is <a href="msscyc_1.html#V5">finitely_operated</a>
 ;<br/>

<b>set </b><font color="Maroon">SF</font> = the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span>;<br/>
<a NAME="E2:8"/><i><font color="Green">E42</font></i>: 
 <a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#G3">MSAlgebra</a>(# <span class="p1">(<span class="default"><a href="msafree.html#K8">FreeSort</a> <font color="Maroon">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="msafree.html#K10">FreeOper</a> <font color="Maroon">X</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="msafree.html#D16">MSAFREE:def 16</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>for <font color="Olive">v</font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font> holds  <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">v</font> :  <a href="msafree2.html#K9">depth</a> <font color="Olive">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> </span>}</span>  is <a href="finset_1.html#V1">finite</a>;<br/>
<a NAME="E3:8"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>;<br/>

<b>set </b><font color="Maroon">dle0</font> = <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> :  <a href="msafree2.html#K9">depth</a> <font color="Olive">t</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 </span>}</span> ;<br/>
<b>set </b><font color="Maroon">d0</font> = <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> :  <a href="msafree2.html#K9">depth</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> 0 </span>}</span> ;<br/>
<a NAME="E1:8_1"/>
 <a href="msualg_2.html#K1">Constants</a> <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span>,<font color="Maroon">v</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i>, <a class="ref" href="msscyc_1.html#T26">MSSCYC_1:26</a></i>;<br/>
<a NAME="E2:8_1"/><b>then </b><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><a href="msafree.html#K12">FreeGen</a> <font color="Maroon">v</font>,<font color="Maroon">X</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="msualg_2.html#K1">Constants</a> <span class="p2">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span>,<font color="Maroon">v</font></span>)</span> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="finset_1.html#T14">FINSET_1:14</a></i>;<br/>
<a NAME="E3:8_1"/><i><font color="Green">E56</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> :  <a href="msafree2.html#K9">depth</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> 0 </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msafree.html#K12">FreeGen</a> <font color="Maroon">v</font>,<font color="Maroon">X</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="msualg_2.html#K1">Constants</a> <span class="p2">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span>,<font color="Maroon">v</font></span>)</span>
 
<b>by </b><i><a class="ref" href="msscyc_1.html#T27">MSSCYC_1:27</a></i>;<br/>
<a NAME="E4:8_1"/>
<span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> :  <a href="msafree2.html#K9">depth</a> <font color="Olive">t</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> :  <a href="msafree2.html#K9">depth</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> 0 </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:8_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> :  <a href="msafree2.html#K9">depth</a> <font color="Olive">t</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">t</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font><b> such that </b><br/><a NAME="E3:8_1_1"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> &amp;  <a href="msafree2.html#K9">depth</a> <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 ;<br/>
<a NAME="E4:8_1_1"/>
 <a href="msafree2.html#K9">depth</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:8_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> :  <a href="msafree2.html#K9">depth</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> 0 </span>}</span> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:8_1"/>
<span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> :  <a href="msafree2.html#K9">depth</a> <font color="Olive">t</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i>, , <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:8"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="xcmplx_0.html#K2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:8_2"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">v</font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font> holds  <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">v</font> :  <a href="msafree2.html#K9">depth</a> <font color="Olive">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 ;<br/>

<b>let </b><font color="Maroon">v</font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>;<br/>

<a NAME="E2:8_2"/><i><font color="Green">E63</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="msafree.html#K7">FreeSort</a> <font color="Maroon">X</font>,<font color="Maroon">v</font>
 
<b>by </b><i>, <a class="ref" href="msafree.html#D13">MSAFREE:def 13</a></i>;<br/>
<a NAME="E3:8_2"/><i><font color="Green">E64</font></i>: 
 <a href="msafree.html#K7">FreeSort</a> <font color="Maroon">X</font>,<font color="Maroon">v</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> <a href="msaterm.html#K1">-Terms</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="msaterm.html#T12">MSATERM:12</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>] <b>means </b> <a href="msafree2.html#K9">depth</a> <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>] <b>means </b> <a href="msafree2.html#K9">depth</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1;<br/>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>] <b>means </b> <a href="msafree2.html#K9">depth</a> <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1;<br/>
<b>defpred </b><font color="Maroon">S<sub>5</sub></font>[   <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>] <b>means </b>(  <a href="msafree2.html#K9">depth</a> <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> or  <a href="msafree2.html#K9">depth</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 );<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">dn1</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">t</font>) where <font color="Olive">t</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">t</font>] </span>}</span> ;<br/>
<b>set </b><font color="Maroon">dn11</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">t</font>) where <font color="Olive">t</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> : <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">t</font>] </span>}</span> ;<br/>
<a NAME="E4:8_2"/><i><font color="Green">E84</font></i>: 
for <font color="Olive">t</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> holds <br/> ( <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">t</font>] iff <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">t</font>] )
 
<b>by </b><i><a class="ref" href="nat_1.html#T26">NAT_1:26</a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E5:8_2"/><i><font color="Green">E85</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">t</font>) where <font color="Olive">t</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">t</font>] </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">t</font>) where <font color="Olive">t</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> : <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">t</font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="fraenkel.html#S3">FRAENKEL:sch 3</a>();<br/></i>
<b>set </b><font color="Maroon">dln</font> = <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">t</font>] </span>}</span> ;<br/>
<b>set </b><font color="Maroon">den1</font> = <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">t</font>] </span>}</span> ;<br/>
<a NAME="E6:8_2"/><i><font color="Green">E86</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> : ( <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">t</font>] or <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">t</font>] ) </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">t</font>] </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">t</font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="topreal1.html#S1">TOPREAL1:sch 1</a>();<br/></i>
<b>set </b><font color="Maroon">ov</font> = <span class="p1">{<span class="default"> <font color="Olive">o</font> where <font color="Olive">o</font> is   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font> :  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> </span>}</span> ;<br/>
<a NAME="E7:8_2"/>
<span class="p1">{<span class="default"> <font color="Olive">o</font> where <font color="Olive">o</font> is   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font> :  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i>, <a class="ref" href="msscyc_1.html#D6">MSSCYC_1:def 6</a></i>;<br/>
<b>then consider </b><font color="Maroon">ovs</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E9:8_2"/><i><font color="Green">E87</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">ovs</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">o</font> where <font color="Olive">o</font> is   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font> :  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T73">FINSEQ_1:73</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> : (  <a href="msafree2.html#K9">depth</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Olive">t</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">ovs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> ) </span>}</span> ;<br/>
<b>consider </b><font color="Maroon">dvs</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E11:8_2"/><i><font color="Green">E88</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">dvs</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">ovs</font> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">ovs</font></span>)</span> holds <br/><font color="Maroon">dvs</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">k</font>) ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S2">FINSEQ_1:sch 2</a>();<br/></i>
<a NAME="E12:8_2"/><i><font color="Green">E89</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">ovs</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">ovs</font></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">dvs</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">dvs</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E13:8_2"/>
for <font color="Olive">k</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">dvs</font> holds <br/><font color="Maroon">dvs</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> is <a href="finset_1.html#V1">finite</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2_1"><b>proof </b></a><div class="add">

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

<b>set </b><font color="Maroon">dvsk</font> = <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> : (  <a href="msafree2.html#K9">depth</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Olive">t</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">ovs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> ) </span>}</span> ;<br/>
<b>assume </b><a NAME="E1:8_2_1"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">dvs</font>
 ;<br/>

<a NAME="E2:8_2_1"/><b>then </b><i><font color="Green">E95</font></i>: 
<font color="Maroon">dvs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> : (  <a href="msafree2.html#K9">depth</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Olive">t</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">ovs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> ) </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:8_2_1"/>
<font color="Maroon">ovs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">ovs</font>
 
<b>by </b><i>, , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">o</font> being   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E5:8_2_1"/><i><font color="Green">E96</font></i>: 
( <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ovs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp;  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">davsk</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Olive">a</font></span>)</span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="msaterm.html#K1">-Terms</a> <font color="Maroon">X</font></span>)</span> <a href="finseq_1.html#K13">*</a>  : ( <font color="Olive">a</font> is    <a href="msaterm.html#M1">ArgumentSeq</a> of  <a href="msaterm.html#K2">Sym</a> <font color="Maroon">o</font>,<font color="Maroon">X</font> &amp;  ex <font color="Olive">t</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> st <br/>(  <a href="msafree2.html#K9">depth</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Olive">a</font> ) ) </span>}</span> ;<br/>
<a NAME="E6:8_2_1"/><i><font color="Green">E97</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> : (  <a href="msafree2.html#K9">depth</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Olive">t</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">ovs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Olive">a</font></span>)</span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="msaterm.html#K1">-Terms</a> <font color="Maroon">X</font></span>)</span> <a href="finseq_1.html#K13">*</a>  : ( <font color="Olive">a</font> is    <a href="msaterm.html#M1">ArgumentSeq</a> of  <a href="msaterm.html#K2">Sym</a> <font color="Maroon">o</font>,<font color="Maroon">X</font> &amp;  ex <font color="Olive">t</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> st <br/>(  <a href="msafree2.html#K9">depth</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Olive">a</font> ) ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:8_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> : (  <a href="msafree2.html#K9">depth</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Olive">t</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">ovs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">t</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font><b> such that </b><br/><a NAME="E3:8_2_1_1"/><i><font color="Green">E99</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> &amp;  <a href="msafree2.html#K9">depth</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i/>;<br/>
<a NAME="E4:8_2_1_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="msafree.html#K7">FreeSort</a> <font color="Maroon">X</font>,<font color="Maroon">v</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">t'</font> = <font color="Maroon">t</font> as   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">X</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">a</font> being    <a href="msaterm.html#M1">ArgumentSeq</a> of  <a href="msaterm.html#K2">Sym</a> <font color="Maroon">o</font>,<font color="Maroon">X</font><b> such that </b><br/><a NAME="E7:8_2_1_1"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">t'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="msaterm.html#T10">MSATERM:10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="msaterm.html#K1">-Terms</a> <font color="Maroon">X</font></span>)</span> <a href="finseq_1.html#K13">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E9:8_2_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">a'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Olive">a</font></span>)</span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="msaterm.html#K1">-Terms</a> <font color="Maroon">X</font></span>)</span> <a href="finseq_1.html#K13">*</a>  : ( <font color="Olive">a</font> is    <a href="msaterm.html#M1">ArgumentSeq</a> of  <a href="msaterm.html#K2">Sym</a> <font color="Maroon">o</font>,<font color="Maroon">X</font> &amp;  ex <font color="Olive">t</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> st <br/>(  <a href="msafree2.html#K9">depth</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Olive">a</font> ) ) </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E10:8_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Olive">a</font></span>)</span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="msaterm.html#K1">-Terms</a> <font color="Maroon">X</font></span>)</span> <a href="finseq_1.html#K13">*</a>  : ( <font color="Olive">a</font> is    <a href="msaterm.html#M1">ArgumentSeq</a> of  <a href="msaterm.html#K2">Sym</a> <font color="Maroon">o</font>,<font color="Maroon">X</font> &amp;  ex <font color="Olive">t</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> st <br/>(  <a href="msafree2.html#K9">depth</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Olive">a</font> ) ) </span>}</span> 
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">avsk</font> = <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="msaterm.html#K1">-Terms</a> <font color="Maroon">X</font></span>)</span> <a href="finseq_1.html#K13">*</a>  : ( <font color="Olive">a</font> is    <a href="msaterm.html#M1">ArgumentSeq</a> of  <a href="msaterm.html#K2">Sym</a> <font color="Maroon">o</font>,<font color="Maroon">X</font> &amp;  ex <font color="Olive">t</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> st <br/>(  <a href="msafree2.html#K9">depth</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Olive">a</font> ) ) </span>}</span> ;<br/>
<a NAME="E7:8_2_1"/><i><font color="Green">E104</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="msaterm.html#K1">-Terms</a> <font color="Maroon">X</font></span>)</span> <a href="finseq_1.html#K13">*</a>  : ( <font color="Olive">a</font> is    <a href="msaterm.html#M1">ArgumentSeq</a> of  <a href="msaterm.html#K2">Sym</a> <font color="Maroon">o</font>,<font color="Maroon">X</font> &amp;  ex <font color="Olive">t</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> st <br/>(  <a href="msafree2.html#K9">depth</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Olive">a</font> ) ) </span>}</span> ,<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Olive">a</font></span>)</span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="msaterm.html#K1">-Terms</a> <font color="Maroon">X</font></span>)</span> <a href="finseq_1.html#K13">*</a>  : ( <font color="Olive">a</font> is    <a href="msaterm.html#M1">ArgumentSeq</a> of  <a href="msaterm.html#K2">Sym</a> <font color="Maroon">o</font>,<font color="Maroon">X</font> &amp;  ex <font color="Olive">t</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> st <br/>(  <a href="msafree2.html#K9">depth</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Olive">a</font> ) ) </span>}</span>  <a href="tarski.html#R2">are_equipotent</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2_1_2"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">Z</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">a</font>,<span class="p3">(<span class="default"><span class="p4"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Olive">a</font></span>)</span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="msaterm.html#K1">-Terms</a> <font color="Maroon">X</font></span>)</span> <a href="finseq_1.html#K13">*</a>  : ( <font color="Olive">a</font> is    <a href="msaterm.html#M1">ArgumentSeq</a> of  <a href="msaterm.html#K2">Sym</a> <font color="Maroon">o</font>,<font color="Maroon">X</font> &amp;  ex <font color="Olive">t</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> st <br/>(  <a href="msafree2.html#K9">depth</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Olive">a</font> ) ) </span>}</span> ;<br/>
<b>take </b>
<span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">a</font>,<span class="p3">(<span class="default"><span class="p4"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Olive">a</font></span>)</span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="msaterm.html#K1">-Terms</a> <font color="Maroon">X</font></span>)</span> <a href="finseq_1.html#K13">*</a>  : ( <font color="Olive">a</font> is    <a href="msaterm.html#M1">ArgumentSeq</a> of  <a href="msaterm.html#K2">Sym</a> <font color="Maroon">o</font>,<font color="Maroon">X</font> &amp;  ex <font color="Olive">t</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> st <br/>(  <a href="msafree2.html#K9">depth</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Olive">a</font> ) ) </span>}</span> 
; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D6">TARSKI:def 6</a><br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:8_2_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="msaterm.html#K1">-Terms</a> <font color="Maroon">X</font></span>)</span> <a href="finseq_1.html#K13">*</a>  : ( <font color="Olive">a</font> is    <a href="msaterm.html#M1">ArgumentSeq</a> of  <a href="msaterm.html#K2">Sym</a> <font color="Maroon">o</font>,<font color="Maroon">X</font> &amp;  ex <font color="Olive">t</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> st <br/>(  <a href="msafree2.html#K9">depth</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Olive">a</font> ) ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">a</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="msaterm.html#K1">-Terms</a> <font color="Maroon">X</font></span>)</span> <a href="finseq_1.html#K13">*</a> <b> such that </b><br/><a NAME="E3:8_2_1_2_1"/><i><font color="Green">E107</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> is    <a href="msaterm.html#M1">ArgumentSeq</a> of  <a href="msaterm.html#K2">Sym</a> <font color="Maroon">o</font>,<font color="Maroon">X</font> &amp;  ex <font color="Olive">t</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> st <br/>(  <a href="msafree2.html#K9">depth</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">a</font> ) )
 ;<br/><b>reconsider </b><font color="Maroon">y'</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">a</font> as    <a href="hidden.html#M1">set</a>  ;<br/><b>take </b><font color="Maroon">y'</font> = <font color="Maroon">y'</font>;<br/><b>thus </b><a NAME="E5:8_2_1_2_1"/>
<font color="Maroon">y'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Olive">a</font></span>)</span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="msaterm.html#K1">-Terms</a> <font color="Maroon">X</font></span>)</span> <a href="finseq_1.html#K13">*</a>  : ( <font color="Olive">a</font> is    <a href="msaterm.html#M1">ArgumentSeq</a> of  <a href="msaterm.html#K2">Sym</a> <font color="Maroon">o</font>,<font color="Maroon">X</font> &amp;  ex <font color="Olive">t</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> st <br/>(  <a href="msafree2.html#K9">depth</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Olive">a</font> ) ) </span>}</span> 
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E6:8_2_1_2_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">a</font>,<span class="p3">(<span class="default"><span class="p4"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Olive">a</font></span>)</span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="msaterm.html#K1">-Terms</a> <font color="Maroon">X</font></span>)</span> <a href="finseq_1.html#K13">*</a>  : ( <font color="Olive">a</font> is    <a href="msaterm.html#M1">ArgumentSeq</a> of  <a href="msaterm.html#K2">Sym</a> <font color="Maroon">o</font>,<font color="Maroon">X</font> &amp;  ex <font color="Olive">t</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> st <br/>(  <a href="msafree2.html#K9">depth</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Olive">a</font> ) ) </span>}</span> 
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:8_2_1_2_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Olive">a</font></span>)</span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="msaterm.html#K1">-Terms</a> <font color="Maroon">X</font></span>)</span> <a href="finseq_1.html#K13">*</a>  : ( <font color="Olive">a</font> is    <a href="msaterm.html#M1">ArgumentSeq</a> of  <a href="msaterm.html#K2">Sym</a> <font color="Maroon">o</font>,<font color="Maroon">X</font> &amp;  ex <font color="Olive">t</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> st <br/>(  <a href="msafree2.html#K9">depth</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Olive">a</font> ) ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">a</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="msaterm.html#K1">-Terms</a> <font color="Maroon">X</font></span>)</span> <a href="finseq_1.html#K13">*</a> <b> such that </b><br/><a NAME="E3:8_2_1_2_2"/><i><font color="Green">E108</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> is    <a href="msaterm.html#M1">ArgumentSeq</a> of  <a href="msaterm.html#K2">Sym</a> <font color="Maroon">o</font>,<font color="Maroon">X</font> &amp;  ex <font color="Olive">t</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> st <br/>(  <a href="msafree2.html#K9">depth</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">a</font> ) )
 ;<br/><b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font> as    <a href="hidden.html#M1">set</a>  ;<br/><b>take </b><font color="Maroon">a'</font> = <font color="Maroon">a'</font>;<br/><b>thus </b><a NAME="E5:8_2_1_2_2"/>
<font color="Maroon">a'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="msaterm.html#K1">-Terms</a> <font color="Maroon">X</font></span>)</span> <a href="finseq_1.html#K13">*</a>  : ( <font color="Olive">a</font> is    <a href="msaterm.html#M1">ArgumentSeq</a> of  <a href="msaterm.html#K2">Sym</a> <font color="Maroon">o</font>,<font color="Maroon">X</font> &amp;  ex <font color="Olive">t</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> st <br/>(  <a href="msafree2.html#K9">depth</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Olive">a</font> ) ) </span>}</span> 
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E6:8_2_1_2_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a'</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">a</font>,<span class="p3">(<span class="default"><span class="p4"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Olive">a</font></span>)</span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="msaterm.html#K1">-Terms</a> <font color="Maroon">X</font></span>)</span> <a href="finseq_1.html#K13">*</a>  : ( <font color="Olive">a</font> is    <a href="msaterm.html#M1">ArgumentSeq</a> of  <a href="msaterm.html#K2">Sym</a> <font color="Maroon">o</font>,<font color="Maroon">X</font> &amp;  ex <font color="Olive">t</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> st <br/>(  <a href="msafree2.html#K9">depth</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Olive">a</font> ) ) </span>}</span> 
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E3:8_2_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">a</font>,<span class="p3">(<span class="default"><span class="p4"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Olive">a</font></span>)</span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="msaterm.html#K1">-Terms</a> <font color="Maroon">X</font></span>)</span> <a href="finseq_1.html#K13">*</a>  : ( <font color="Olive">a</font> is    <a href="msaterm.html#M1">ArgumentSeq</a> of  <a href="msaterm.html#K2">Sym</a> <font color="Maroon">o</font>,<font color="Maroon">X</font> &amp;  ex <font color="Olive">t</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> st <br/>(  <a href="msafree2.html#K9">depth</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Olive">a</font> ) ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">xa</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="msaterm.html#K1">-Terms</a> <font color="Maroon">X</font></span>)</span> <a href="finseq_1.html#K13">*</a> <b> such that </b><br/><a NAME="E5:8_2_1_2"/><i><font color="Green">E109</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">xa</font>,<span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">xa</font></span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">xa</font> is    <a href="msaterm.html#M1">ArgumentSeq</a> of  <a href="msaterm.html#K2">Sym</a> <font color="Maroon">o</font>,<font color="Maroon">X</font> &amp;  ex <font color="Olive">t</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> st <br/>(  <a href="msafree2.html#K9">depth</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">xa</font> ) )
 ;<br/>
<b>assume </b><a NAME="E6:8_2_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">u</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">a</font>,<span class="p3">(<span class="default"><span class="p4"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Olive">a</font></span>)</span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="msaterm.html#K1">-Terms</a> <font color="Maroon">X</font></span>)</span> <a href="finseq_1.html#K13">*</a>  : ( <font color="Olive">a</font> is    <a href="msaterm.html#M1">ArgumentSeq</a> of  <a href="msaterm.html#K2">Sym</a> <font color="Maroon">o</font>,<font color="Maroon">X</font> &amp;  ex <font color="Olive">t</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> st <br/>(  <a href="msafree2.html#K9">depth</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Olive">a</font> ) ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">za</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="msaterm.html#K1">-Terms</a> <font color="Maroon">X</font></span>)</span> <a href="finseq_1.html#K13">*</a> <b> such that </b><br/><a NAME="E8:8_2_1_2"/><i><font color="Green">E110</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">u</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">za</font>,<span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">za</font></span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">za</font> is    <a href="msaterm.html#M1">ArgumentSeq</a> of  <a href="msaterm.html#K2">Sym</a> <font color="Maroon">o</font>,<font color="Maroon">X</font> &amp;  ex <font color="Olive">t</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> st <br/>(  <a href="msafree2.html#K9">depth</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">za</font> ) )
 ;<br/>
<a NAME="E9:8_2_1_2"/><i><font color="Green">E111</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xa</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">xa</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">za</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">za</font> )
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E10:8_2_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> implies <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> )
 ;<br/>

<b>assume </b><a NAME="E11:8_2_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font>
 ;<br/>

<b>hence </b><a NAME="E12:8_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>by </b><i>, , , <a class="ref" href="trees_4.html#T15">TREES_4:15</a></i>;<br/>


</div><b>end;</b></div>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> :  <a href="msafree2.html#K9">depth</a> <font color="Olive">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> </span>}</span> ;<br/>
<b>consider </b><font color="Maroon">nS</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E9:8_2_1"/><i><font color="Green">E112</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">nS</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> holds <br/><font color="Maroon">nS</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">k</font>) ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S2">FINSEQ_1:sch 2</a>();<br/></i>
<a NAME="E10:8_2_1"/><i><font color="Green">E113</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">nS</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">nS</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:8_2_1_3"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">nS</font>
 ;<br/><b>then reconsider </b><font color="Maroon">k</font> = <font color="Maroon">x</font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>set </b><font color="Maroon">nSk</font> = <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span> :  <a href="msafree2.html#K9">depth</a> <font color="Olive">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> </span>}</span> ;<br/><a NAME="E3:8_2_1_3"/>
<font color="Maroon">nS</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span> :  <a href="msafree2.html#K9">depth</a> <font color="Olive">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> </span>}</span> 
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E4:8_2_1_3"/>
<font color="Maroon">nS</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E12:8_2_1"/><b>then </b><i><font color="Green">E115</font></i>: 
 <a href="card_3.html#K4">product</a> <font color="Maroon">nS</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="msscyc_1.html#T1">MSSCYC_1:1</a></i>;<br/>
<a NAME="E13:8_2_1"/>
<span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="msaterm.html#K1">-Terms</a> <font color="Maroon">X</font></span>)</span> <a href="finseq_1.html#K13">*</a>  : ( <font color="Olive">a</font> is    <a href="msaterm.html#M1">ArgumentSeq</a> of  <a href="msaterm.html#K2">Sym</a> <font color="Maroon">o</font>,<font color="Maroon">X</font> &amp;  ex <font color="Olive">t</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> st <br/>(  <a href="msafree2.html#K9">depth</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Olive">a</font> ) ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">nS</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2_1_4"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:8_2_1_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="msaterm.html#K1">-Terms</a> <font color="Maroon">X</font></span>)</span> <a href="finseq_1.html#K13">*</a>  : ( <font color="Olive">a</font> is    <a href="msaterm.html#M1">ArgumentSeq</a> of  <a href="msaterm.html#K2">Sym</a> <font color="Maroon">o</font>,<font color="Maroon">X</font> &amp;  ex <font color="Olive">t</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> st <br/>(  <a href="msafree2.html#K9">depth</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Olive">a</font> ) ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="msaterm.html#K1">-Terms</a> <font color="Maroon">X</font></span>)</span> <a href="finseq_1.html#K13">*</a> <b> such that </b><br/><a NAME="E3:8_2_1_4"/><i><font color="Green">E116</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> is    <a href="msaterm.html#M1">ArgumentSeq</a> of  <a href="msaterm.html#K2">Sym</a> <font color="Maroon">o</font>,<font color="Maroon">X</font> &amp;  ex <font color="Olive">t</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> st <br/>(  <a href="msafree2.html#K9">depth</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">a</font> ) )
 ;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as    <a href="msaterm.html#M1">ArgumentSeq</a> of  <a href="msaterm.html#K2">Sym</a> <font color="Maroon">o</font>,<font color="Maroon">X</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">t</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font><b> such that </b><br/><a NAME="E6:8_2_1_4"/><i><font color="Green">E117</font></i>: 
(  <a href="msafree2.html#K9">depth</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">a</font> )
 <b>by </b><i/>;<br/>
<a NAME="E7:8_2_1_4"/><i><font color="Green">E118</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="msaterm.html#T22">MSATERM:22</a></i>;<br/>
<a NAME="E8:8_2_1_4"/><i><font color="Green">E119</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2_1_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:8_2_1_4_1"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a</font>
 ;<br/><b>then reconsider </b><font color="Maroon">k</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><b>reconsider </b><font color="Maroon">ak</font> = <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> as   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">X</font> <b>by </b><i>, <a class="ref" href="msaterm.html#T22">MSATERM:22</a></i>;<br/><a NAME="E4:8_2_1_4_1"/><i><font color="Green">E121</font></i>: 
( <font color="Maroon">ak</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> &amp;  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">ak</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp;  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">ak</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> )
 <b>by </b><i>, <a class="ref" href="msaterm.html#T23">MSATERM:23</a></i>;<br/><a NAME="E5:8_2_1_4_1"/>
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">ak</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="msafree.html#K7">FreeSort</a> <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">ak</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="msafree.html#D13">MSAFREE:def 13</a></i>;<br/><b>then reconsider </b><font color="Maroon">ak</font> = <font color="Maroon">ak</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span> <b>by </b><i>, <a class="ref" href="msaterm.html#D5">MSATERM:def 5</a></i>;<br/><b>set </b><font color="Maroon">nSk</font> = <span class="p1">{<span class="default"> <font color="Olive">tk</font> where <font color="Olive">tk</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span> :  <a href="msafree2.html#K9">depth</a> <font color="Olive">tk</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> </span>}</span> ;<br/><a NAME="E7:8_2_1_4_1"/><i><font color="Green">E124</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">tk</font> where <font color="Olive">tk</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span> :  <a href="msafree2.html#K9">depth</a> <font color="Olive">tk</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> </span>}</span>  <a href="hidden.html#R1">=</a> <font color="Maroon">nS</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , , </i>;<br/><a NAME="E8:8_2_1_4_1"/>
 <a href="msafree2.html#K9">depth</a> <font color="Maroon">ak</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i>, , <a class="ref" href="msscyc_1.html#T28">MSSCYC_1:28</a></i>;<br/><a NAME="E9:8_2_1_4_1"/><b>then </b>
 <a href="msafree2.html#K9">depth</a> <font color="Maroon">ak</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E10:8_2_1_4_1"/>
<font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">nS</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:8_2_1_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">nS</font>
 <b>by </b><i>, , , , , <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:8_2_1"/><b>then </b>
<span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="msaterm.html#K1">-Terms</a> <font color="Maroon">X</font></span>)</span> <a href="finseq_1.html#K13">*</a>  : ( <font color="Olive">a</font> is    <a href="msaterm.html#M1">ArgumentSeq</a> of  <a href="msaterm.html#K2">Sym</a> <font color="Maroon">o</font>,<font color="Maroon">X</font> &amp;  ex <font color="Olive">t</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> st <br/>(  <a href="msafree2.html#K9">depth</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Olive">a</font> ) ) </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<a NAME="E15:8_2_1"/><b>then </b>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Olive">a</font></span>)</span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="msaterm.html#K1">-Terms</a> <font color="Maroon">X</font></span>)</span> <a href="finseq_1.html#K13">*</a>  : ( <font color="Olive">a</font> is    <a href="msaterm.html#M1">ArgumentSeq</a> of  <a href="msaterm.html#K2">Sym</a> <font color="Maroon">o</font>,<font color="Maroon">X</font> &amp;  ex <font color="Olive">t</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> st <br/>(  <a href="msafree2.html#K9">depth</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Olive">a</font> ) ) </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i>, <a class="ref" href="card_1.html#T68">CARD_1:68</a></i>;<br/>
<b>hence </b><a NAME="E16:8_2_1"/>
<font color="Maroon">dvs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i>, , <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:8_2"/><b>then </b><i><font color="Green">E126</font></i>: 
 <a href="card_3.html#K3">Union</a> <font color="Maroon">dvs</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="card_4.html#T63">CARD_4:63</a></i>;<br/>
<a NAME="E15:8_2"/>
<span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">t</font>] </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K3">Union</a> <font color="Maroon">dvs</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:8_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">t</font>] </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">t</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font><b> such that </b><br/><a NAME="E3:8_2_2"/><i><font color="Green">E127</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> &amp;  <a href="msafree2.html#K9">depth</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 )
 ;<br/>
<a NAME="E4:8_2_2"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="msafree.html#K7">FreeSort</a> <font color="Maroon">X</font>,<font color="Maroon">v</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">t'</font> = <font color="Maroon">t</font> as   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">X</font> <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:8_2_2_1"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">t'</font> is <a href="trees_9.html#V1">root</a>
 ;<br/><b>consider </b><font color="Maroon">dt</font> being  <a href="finset_1.html#V1">finite</a> <a href="trees_2.html#NM4">DecoratedTree</a>, <font color="Maroon">tt</font> being  <a href="finset_1.html#V1">finite</a> <a href="trees_1.html#NM1">Tree</a><b> such that </b><br/><a NAME="E3:8_2_2_1"/><i><font color="Green">E129</font></i>: 
( <font color="Maroon">dt</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> &amp; <font color="Maroon">tt</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">dt</font> &amp;  <a href="msafree2.html#K9">depth</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="trees_1.html#K6">height</a> <font color="Maroon">tt</font> )
 <b>by </b><i><a class="ref" href="msafree2.html#D14">MSAFREE2:def 14</a></i>;<br/><b>thus </b><a NAME="E4:8_2_2_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="trees_1.html#T79">TREES_1:79</a>, <a class="ref" href="trees_9.html#D1">TREES_9:def 1</a></i>;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon">o</font> being   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E8:8_2_2"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">t'</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="msscyc_1.html#T20">MSSCYC_1:20</a></i>;<br/>
 <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font> = 
 <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t'</font>
<b>by </b><i>, <a class="ref" href="msaterm.html#T17">MSATERM:17</a></i>
<br/>.= 
<font color="Maroon">v</font>
<b>by </b><i>, <a class="ref" href="msaterm.html#D5">MSATERM:def 5</a></i>
;<br/>
<a NAME="E10:8_2_2"/><b>then </b>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">ovs</font>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">k</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:8_2_2"/><i><font color="Green">E131</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">ovs</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ovs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E13:8_2_2"/>
<font color="Maroon">dvs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">t1</font> where <font color="Olive">t1</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> : (  <a href="msafree2.html#K9">depth</a> <font color="Olive">t1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Olive">t1</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">ovs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> ) </span>}</span> 
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E14:8_2_2"/><b>then </b><i><font color="Green">E132</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">dvs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E15:8_2_2"/>
<font color="Maroon">dvs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">dvs</font>
 
<b>by </b><i>, , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E16:8_2_2"/><b>then </b>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">dvs</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E17:8_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K3">Union</a> <font color="Maroon">dvs</font>
 <b>by </b><i>, <a class="ref" href="card_3.html#D4">CARD_3:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:8_2"/><b>then </b><i><font color="Green">E134</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">t</font>] </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<a NAME="E17:8_2"/>
<span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">t</font>] </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E18:8_2"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">t</font>) where <font color="Olive">t</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">t</font>] </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i>, , , <a class="ref" href="finset_1.html#T14">FINSET_1:14</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E5:8"/>
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(, );</i><br/>


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