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

<b>set </b><font color="Maroon">c<sub>1</sub></font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="pboole.html#K2">\/</a> <font color="Maroon">F<sub>3</sub></font>();<br/>
<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="msafree.html#K5">DTConMSA</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="pboole.html#K2">\/</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span>;<br/>
<a NAME="E1:16_1"/><i><font color="Green">E12</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="lang1.html#NM1">Symbol</a> of <span class="p1">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> <span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="pboole.html#K2">\/</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="dtconstr.html#K6">Terminals</a> <span class="p1">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> <span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="pboole.html#K2">\/</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span></span>)</span> holds <br/><font color="Maroon">P<sub>1</sub></font>[ <a href="trees_4.html#K2">root-tree</a> <font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="lang1.html#NM1">Symbol</a> of <span class="p1">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> <span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="pboole.html#K2">\/</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span></span>)</span>;<br/>

<b>assume </b><a NAME="E1:16_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="dtconstr.html#K6">Terminals</a> <span class="p1">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> <span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="pboole.html#K2">\/</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:16_1_1"/><b>then </b>
( ex <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>()ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> ) or ex <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>()ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> st <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> )
 
<b>by </b><i><a class="txt" href="msaterm.html#E7"><i><font color="Green">Lemma7</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:16_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[ <a href="trees_4.html#K2">root-tree</a> <font color="Maroon">c<sub>3</sub></font>]
 <b>by </b><i><a class="txt" href="msaterm.html#E1:16"><i><font color="Green">E9</font></i></a>, <a class="txt" href="msaterm.html#E2:16"><i><font color="Green">E10</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:16_1"/><i><font color="Green">E13</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="lang1.html#NM1">Symbol</a> of <span class="p1">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> <span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="pboole.html#K2">\/</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span></span>)</span><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="trees_4.html#M1">FinSequence</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> <span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="pboole.html#K2">\/</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="lang1.html#R1">==&gt;</a>  <a href="dtconstr.html#K1">roots</a> <font color="Olive">b<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being  <a href="trees_2.html#NM6">DecoratedTree</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> <span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="pboole.html#K2">\/</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span></span>)</span> st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>3</sub></font>] ) holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="trees_4.html#K12">-tree</a> <font color="Olive">b<sub>2</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="lang1.html#NM1">Symbol</a> of <span class="p1">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> <span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="pboole.html#K2">\/</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="trees_4.html#M1">FinSequence</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> <span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="pboole.html#K2">\/</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span></span>)</span>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:16_1_2"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="lang1.html#R1">==&gt;</a>  <a href="dtconstr.html#K1">roots</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E2:16_1_2"/><i><font color="Green">E15</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="trees_2.html#NM6">DecoratedTree</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> <span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="pboole.html#K2">\/</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 ;<br/>

<a NAME="E3:16_1_2"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="lang1.html#NM1">Symbol</a> of <span class="p1">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> <span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="pboole.html#K2">\/</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span></span>)</span> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <font color="Olive">b<sub>1</sub></font> <a href="lang1.html#R1">==&gt;</a> <font color="Olive">b<sub>2</sub></font> </span>}</span>  &amp;  <a href="dtconstr.html#K7">NonTerminals</a> <span class="p1">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> <span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="pboole.html#K2">\/</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="lang1.html#NM1">Symbol</a> of <span class="p1">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> <span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="pboole.html#K2">\/</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span></span>)</span> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <font color="Olive">b<sub>1</sub></font> <a href="lang1.html#R1">==&gt;</a> <font color="Olive">b<sub>2</sub></font> </span>}</span>  )
 
<b>by </b><i><a class="txt" href="msaterm.html#E1:16_1_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="lang1.html#D3">LANG1:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="dtconstr.html#NM3">NonTerminal</a> of <span class="p1">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> <span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="pboole.html#K2">\/</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span></span>)</span> ;<br/>
<a NAME="E5:16_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">F<sub>1</sub></font>(),<span class="p2"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()</span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="msaterm.html#E8"><i><font color="Green">Lemma8</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">c<sub>7</sub></font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()</span><a href="tarski.html#K1">}</a></span><b> such that </b><br/><a NAME="E7:16_1_2"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/>
<a NAME="E8:16_1_2"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="trees_4.html#M1">FinSequence</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="msaterm.html#K1" target="_self">-Terms</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="pboole.html#K2">\/</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span> ;<br/>
<a NAME="E10:16_1_2"/><i><font color="Green">E18</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a>  <a href="msaterm.html#K2" target="_self">Sym</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="pboole.html#K2">\/</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span>
 
<b>by </b><i><a class="ref" href="msafree.html#D11">MSAFREE:def 11</a></i>;<br/>
<a NAME="E11:16_1_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> is    <a href="dtconstr.html#M2">SubtreeSeq</a> of  <a href="msaterm.html#K2" target="_self">Sym</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="pboole.html#K2">\/</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span>
 
<b>by </b><i><a class="txt" href="msaterm.html#E1:16_1_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="msaterm.html#E7:16_1_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="msaterm.html#E8:16_1_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="dtconstr.html#D9">DTCONSTR:def 9</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as    <a href="msaterm.html#M1" target="_self">ArgumentSeq</a> of  <a href="msaterm.html#K2" target="_self">Sym</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="pboole.html#K2">\/</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span> <b>by </b><i><a class="ref" href="msaterm.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E13:16_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="msaterm.html#NM3">c-Term</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>() st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>by </b><i><a class="txt" href="msaterm.html#E2:16_1_2"><i><font color="Green">E15</font></i></a></i>;<br/>
<b>hence </b><a NAME="E14:16_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">c<sub>4</sub></font>]
 <b>by </b><i><a class="txt" href="msaterm.html#E3:16"><i><font color="Green">E11</font></i></a>, <a class="txt" href="msaterm.html#E7:16_1_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="msaterm.html#E8:16_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="msaterm.html#E10:16_1_2"><i><font color="Green">E18</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:16_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="trees_2.html#NM6">DecoratedTree</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> <span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="pboole.html#K2">\/</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> <span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="pboole.html#K2">\/</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span></span>)</span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="dtconstr.html#S7">DTCONSTR:sch 7</a>(<a class="txt" href="msaterm.html#E1:16_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="msaterm.html#E2:16_1"><i><font color="Green">E13</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:16_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="msaterm.html#NM3">c-Term</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>() holds <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 ;<br/>


</div>
