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

<b>let </b><font color="Maroon">S</font> be  non <a href="msualg_1.html#V2">void</a> <a href="catalg_1.html#NM1">Signature</a>;<br/><b>let </b><font color="Maroon">X</font> be  <a href="relat_1.html#V2">V3</a>  <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>, <font color="Maroon">Y</font> be  <a href="relat_1.html#V2">V3</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>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="trees_2.html#NM4">DecoratedTree</a>] <b>means </b>( <font color="Maroon">Y</font> <a href="msafree3.html#K3" target="_self">variables_in</a> <font color="Maroon">a<sub>1</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">X</font> implies <font color="Maroon">a<sub>1</sub></font> is   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">X</font> );<br/>
<a NAME="E1:42"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">s</font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font><br/> for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font> holds  <font color="Maroon">S<sub>1</sub></font>[ <a href="trees_4.html#K2">root-tree</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">s</font></span><a href="domain_1.html#K1">]</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>;<br/>



<b>assume </b><a NAME="E1:42_1"/>
<font color="Maroon">Y</font> <a href="msafree3.html#K3" target="_self">variables_in</a> <span class="p1">(<span class="default"><a href="trees_4.html#K2">root-tree</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="pboole.html#R2">c=</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:42_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="msafree3.html#K3" target="_self">variables_in</a> <span class="p2">(<span class="default"><a href="trees_4.html#K2">root-tree</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="msafree3.html#K3" target="_self">variables_in</a> <span class="p2">(<span class="default"><a href="trees_4.html#K2">root-tree</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> )
 
<b>by </b><i>, <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>
<a NAME="E3:42_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<b>hence </b><a NAME="E4:42_1"/>
 <a href="trees_4.html#K2">root-tree</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span> is   <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#T4">MSATERM:4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:42"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">o</font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font><br/> for <font color="Olive">p</font> being   <a href="msaterm.html#M1">ArgumentSeq</a> of  <a href="msaterm.html#K2">Sym</a> <font color="Olive">o</font>,<font color="Maroon">Y</font>  st ( for <font color="Olive">t</font> being  <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">Y</font>  st <font color="Olive">t</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Olive">p</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">t</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">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">p</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">o</font> be   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">p</font> be    <a href="msaterm.html#M1">ArgumentSeq</a> of  <a href="msaterm.html#K2">Sym</a> <font color="Maroon">o</font>,<font color="Maroon">Y</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:42_2"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">t</font> being  <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">Y</font>  st <font color="Olive">t</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">p</font> &amp; <font color="Maroon">Y</font> <a href="msafree3.html#K3" target="_self">variables_in</a> <font color="Olive">t</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">X</font> holds <br/><font color="Olive">t</font> is   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E2:42_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">Y</font> <a href="msafree3.html#K3" target="_self">variables_in</a> <span class="p1">(<span class="default"><span class="p2"><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">p</font></span>)</span> <a href="pboole.html#R2">c=</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E3:42_2"/><i><font color="Green">E50</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</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>
 
<b>by </b><i><a class="ref" href="msaterm.html#T22">MSATERM:22</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:42_2_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font>
 ;<br/><b>then reconsider </b><font color="Maroon">t</font> = <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">Y</font> <b>by </b><i><a class="ref" href="msaterm.html#T22">MSATERM:22</a></i>;<br/><a NAME="E3:42_2_1"/><i><font color="Green">E64</font></i>: 
(  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t</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">i</font> &amp; <font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">p</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, <a class="ref" href="msaterm.html#T23">MSATERM:23</a></i>;<br/><a NAME="E4:42_2_1"/>
<font color="Maroon">Y</font> <a href="msafree3.html#K3" target="_self">variables_in</a> <font color="Maroon">t</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a><br/>

<b>assume </b><a NAME="E1:42_2_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">s</font> = <font color="Maroon">y</font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font> ;<br/>
<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="E3:42_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="msafree3.html#K3" target="_self">variables_in</a> <font color="Maroon">t</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E4:42_2_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="msafree3.html#K3" target="_self">variables_in</a> <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">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="msafree3.html#K3" target="_self">variables_in</a> <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">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> )
 
<b>by </b><i>, , , <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>
<b>hence </b><a NAME="E5:42_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 ;<br/>


</div><b>end;</b></div><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>, </i>;<br/><b>take </b><font color="Maroon">t'</font> = <font color="Maroon">t'</font>;<br/><b>thus </b><a NAME="E6:42_2_1"/>
<font color="Maroon">t'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/><b>thus </b><a NAME="E7:42_2_1"/>
 <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t'</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">i</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</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>, <a class="ref" href="msaterm.html#T24">MSATERM:24</a></i>;<br/>
<a NAME="E6:42_2"/>
<span class="p1">(<span class="default"><a href="msaterm.html#K2">Sym</a> <font color="Maroon">o</font>,<font color="Maroon">X</font></span>)</span> <a href="msaterm.html#K6">-tree</a> <font color="Maroon">p</font> is   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">X</font>
 
;<br/>
<b>hence </b><a NAME="E7:42_2"/>
<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">p</font> is   <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="msafree.html#D11">MSAFREE:def 11</a></i>;<br/>


</div><b>end;</b></div>
<b>let </b><font color="Maroon">t</font> be   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">Y</font>;<br/>

<b>assume </b><a NAME="E3:42"/>
 <a href="msafree3.html#K4" target="_self">variables_in</a> <font color="Maroon">t</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E4:42"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">Y</font> <a href="msafree3.html#K3" target="_self">variables_in</a> <font color="Maroon">t</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:42"/>
for <font color="Olive">t</font> being  <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">Y</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">t</font>]
 
<b>from </b><i><a class="ref" href="msaterm.html#S1">MSATERM:sch 1</a>(, );<br/></i>
<b>hence </b><a NAME="E6:42"/>
<font color="Maroon">t</font> is   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">X</font>
 <b>by </b><i/>;<br/>


</div>
