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<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">A</font> be  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">V</font> be    <a href="msaterm.html#M4" target="_self">Variables</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">t</font> be   <a href="msaterm.html#NM3">c-Term</a> of <font color="Maroon">A</font>,<font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">f</font> be    <a href="pboole.html#M3">ManySortedFunction</a> of <font color="Maroon">V</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font>;<br/>









<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="msaterm.html#NM3">c-Term</a> of <font color="Maroon">A</font>,<font color="Maroon">V</font>] <b>means </b>for <font color="Olive">vt</font> being <a href="finset_1.html#V1">finite</a> <a href="trees_2.html#NM4">DecoratedTree</a>  st <font color="Olive">vt</font> <a href="msaterm.html#R1" target="_self">is_an_evaluation_of</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">f</font> holds <br/><font color="Olive">vt</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msaterm.html#K7" target="_self">the_sort_of</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<div><i><font color="Green">E26</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><b>now </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 the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>;<br/><b>thus </b><a NAME="E1:66_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font> <a href="msaterm.html#K4" target="_self">-term</a> <font color="Maroon">A</font>,<font color="Maroon">V</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">t</font> = <font color="Maroon">x</font> <a href="msaterm.html#K4" target="_self">-term</a> <font color="Maroon">A</font>,<font color="Maroon">V</font>;<br/>
<b>let </b><font color="Maroon">vt</font> be  <a href="finset_1.html#V1">finite</a> <a href="trees_2.html#NM4">DecoratedTree</a>;<br/>

<b>assume </b><a NAME="E1:66_1_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">vt</font> <a href="msaterm.html#R1" target="_self">is_an_evaluation_of</a> <font color="Maroon">x</font> <a href="msaterm.html#K4" target="_self">-term</a> <font color="Maroon">A</font>,<font color="Maroon">V</font>,<font color="Maroon">f</font>
 ;<br/>

<a NAME="E2:66_1_1"/>
 <a href="trees_4.html#K2">root-tree</a> <font color="Maroon">x</font> <a href="msaterm.html#R1" target="_self">is_an_evaluation_of</a> <font color="Maroon">x</font> <a href="msaterm.html#K4" target="_self">-term</a> <font color="Maroon">A</font>,<font color="Maroon">V</font>,<font color="Maroon">f</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:66_1_1"/><b>then </b>
<font color="Maroon">vt</font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K2">root-tree</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="msaterm.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E4:66_1_1"/><b>then </b>
( <font color="Maroon">vt</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="msaterm.html#K7" target="_self">the_sort_of</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="msaterm.html#K4" target="_self">-term</a> <font color="Maroon">A</font>,<font color="Maroon">V</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="msaterm.html#E1"><i><font color="Green">Lemma22</font></i></a>, <a class="ref" href="trees_4.html#T3">TREES_4:3</a></i>;<br/>
<b>hence </b><a NAME="E5:66_1_1"/>
<font color="Maroon">vt</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msaterm.html#K7" target="_self">the_sort_of</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="msaterm.html#K4" target="_self">-term</a> <font color="Maroon">A</font>,<font color="Maroon">V</font></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<div><i><font color="Green">E28</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2"><b>now </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">v</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>;<br/><b>thus </b><a NAME="E1:66_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">v</font> <a href="msaterm.html#K5" target="_self">-term</a> <font color="Maroon">A</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">t</font> = <font color="Maroon">v</font> <a href="msaterm.html#K5" target="_self">-term</a> <font color="Maroon">A</font>;<br/>
<b>let </b><font color="Maroon">vt</font> be  <a href="finset_1.html#V1">finite</a> <a href="trees_2.html#NM4">DecoratedTree</a>;<br/>

<b>assume </b><a NAME="E1:66_2_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">vt</font> <a href="msaterm.html#R1" target="_self">is_an_evaluation_of</a> <font color="Maroon">v</font> <a href="msaterm.html#K5" target="_self">-term</a> <font color="Maroon">A</font>,<font color="Maroon">f</font>
 ;<br/>

<a NAME="E2:66_2_1"/>
 <a href="trees_4.html#K2">root-tree</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span> <a href="msaterm.html#R1" target="_self">is_an_evaluation_of</a> <font color="Maroon">v</font> <a href="msaterm.html#K5" target="_self">-term</a> <font color="Maroon">A</font>,<font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="msaterm.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E3:66_2_1"/><b>then </b>
<font color="Maroon">vt</font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K2">root-tree</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span>
 
<b>by </b><i><a class="txt" href="msaterm.html#E1:66_2_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="msaterm.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E4:66_2_1"/><b>then </b>
( <font color="Maroon">vt</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> &amp; <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="msaterm.html#K7" target="_self">the_sort_of</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="msaterm.html#K5" target="_self">-term</a> <font color="Maroon">A</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="trees_4.html#T3">TREES_4:3</a></i>;<br/>
<b>hence </b><a NAME="E5:66_2_1"/>
<font color="Maroon">vt</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msaterm.html#K7" target="_self">the_sort_of</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="msaterm.html#K5" target="_self">-term</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<div><i><font color="Green">E51</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="66_3"><b>now </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#NM4">ArgumentSeq</a> of <font color="Maroon">o</font>,<font color="Maroon">A</font>,<font color="Maroon">V</font>;<br/><b>assume </b><a NAME="E1:66_3"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">t</font> being  <a href="msaterm.html#NM3">c-Term</a> of <font color="Maroon">A</font>,<font color="Maroon">V</font>  st <font color="Olive">t</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">t</font>]
 ;<br/><b>thus </b><a NAME="E2:66_3"/>
<font color="Maroon">S<sub>1</sub></font>[<span class="p1">(<span class="default"><a href="msaterm.html#K2" target="_self">Sym</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="pboole.html#K2">\/</a> <font color="Maroon">V</font></span>)</span></span>)</span> <a href="msaterm.html#K6" target="_self">-tree</a> <font color="Maroon">p</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_3_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">t</font> = <span class="p1">(<span class="default"><a href="msaterm.html#K2" target="_self">Sym</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="pboole.html#K2">\/</a> <font color="Maroon">V</font></span>)</span></span>)</span> <a href="msaterm.html#K6" target="_self">-tree</a> <font color="Maroon">p</font>;<br/>
<b>let </b><font color="Maroon">vt</font> be  <a href="finset_1.html#V1">finite</a> <a href="trees_2.html#NM4">DecoratedTree</a>;<br/>

<b>assume </b><a NAME="E1:66_3_1"/>
<font color="Maroon">vt</font> <a href="msaterm.html#R1" target="_self">is_an_evaluation_of</a> <span class="p1">(<span class="default"><a href="msaterm.html#K2" target="_self">Sym</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="pboole.html#K2">\/</a> <font color="Maroon">V</font></span>)</span></span>)</span> <a href="msaterm.html#K6" target="_self">-tree</a> <font color="Maroon">p</font>,<font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">q</font> being  <a href="trees_3.html#V6">DTree-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E3:66_3_1"/><i><font color="Green">E53</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font>
 <b>and </b><br/><a NAME="E4:66_3_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">vt</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="trees_3.html#K16">roots</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">q</font>
 <b>and </b><br/><a NAME="E5:66_3_1"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a><br/> for <font color="Olive">t</font> being  <a href="msaterm.html#NM3">c-Term</a> of <font color="Maroon">A</font>,<font color="Maroon">V</font>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> &amp; <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> holds <br/> ex <font color="Olive">vt</font> being <a href="finset_1.html#V1">finite</a> <a href="trees_2.html#NM4">DecoratedTree</a> st <br/>( <font color="Olive">vt</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">vt</font> <a href="msaterm.html#R1" target="_self">is_an_evaluation_of</a> <font color="Olive">t</font>,<font color="Maroon">f</font> )
 <b>by </b><i/>;<br/>
<a NAME="E6:66_3_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">vt</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="trees_3.html#K16">roots</a> <font color="Maroon">q</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="trees_4.html#D4">TREES_4:def 4</a></i>;<br/>
<a NAME="E7:66_3_1"/><i><font color="Green">E86</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<i><font color="Green">E87</font></i>:  <a href="msualg_1.html#K4">Result</a> <font color="Maroon">o</font>,<font color="Maroon">A</font> = 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
<b>by </b><i><a class="ref" href="msualg_1.html#D10">MSUALG_1:def 10</a></i>
<br/>.= 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font></span>)</span>
<b>by </b><i><a class="txt" href="msaterm.html#E3:66"><i><font color="Green">E51</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span>
<b>by </b><i><a class="ref" href="msualg_1.html#D7">MSUALG_1:def 7</a></i>
<br/>.= 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msaterm.html#K7" target="_self">the_sort_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msaterm.html#K2" target="_self">Sym</a> <font color="Maroon">o</font>,<span class="p4">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="pboole.html#K2">\/</a> <font color="Maroon">V</font></span>)</span></span>)</span> <a href="msaterm.html#K6" target="_self">-tree</a> <font color="Maroon">p</font></span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_3_1_2"><b>now </b></a><div class="add"><a NAME="E1:66_3_1_2"/><i><font color="Green">E116</font></i>: 
(  <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp;  <a href="relat_1.html#K2">rng</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="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E2:66_3_1_2"/><i><font color="Green">E118</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="trees_3.html#K16">roots</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="trees_3.html#D18">TREES_3:def 18</a></i>;<br/><b>hence </b><i><font color="Green">E119</font></i>:  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="trees_3.html#K16">roots</a> <font color="Maroon">q</font></span>)</span> = 
 <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">q</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <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>
<b>by </b><i>, <a class="txt" href="msaterm.html#E1:66"><i><font color="Green">E26</font></i></a></i>
<br/>.= 
 <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="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#NK3" target="_self">*</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>
<b>by </b><i><a class="txt" href="msaterm.html#E3:66_3_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>
;<br/><br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E4:66_3_1_2"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#NK3" target="_self">*</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>
 ;<br/><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:66_3_1_2"/><i><font color="Green">E121</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font> )
 <b>by </b><i><a class="txt" href="msaterm.html#E4:66_3_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="msaterm.html#E8"><i><font color="Green">Lemma55</font></i></a>, </i>;<br/><b>consider </b><font color="Maroon">T</font> being   <a href="trees_2.html#NM4">DecoratedTree</a><b> such that </b><br/><a NAME="E8:66_3_1_2"/><i><font color="Green">E122</font></i>: 
( <font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><a href="trees_3.html#K16">roots</a> <font color="Maroon">q</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="txt" href="msaterm.html#E4:66_3_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="msaterm.html#E8"><i><font color="Green">Lemma55</font></i></a>, , , <a class="ref" href="trees_3.html#D18">TREES_3:def 18</a></i>;<br/><b>reconsider </b><font color="Maroon">t</font> = <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="msaterm.html#NM3">c-Term</a> of <font color="Maroon">A</font>,<font color="Maroon">V</font> <b>by </b><i>, , </i>;<br/><a NAME="E10:66_3_1_2"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font>
 <b>by </b><i>, , </i>;<br/><b>then consider </b><font color="Maroon">vt</font> being  <a href="finset_1.html#V1">finite</a> <a href="trees_2.html#NM4">DecoratedTree</a><b> such that </b><br/><a NAME="E12:66_3_1_2"/><i><font color="Green">E124</font></i>: 
( <font color="Maroon">vt</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">vt</font> <a href="msaterm.html#R1" target="_self">is_an_evaluation_of</a> <font color="Maroon">t</font>,<font color="Maroon">f</font> )
 <b>by </b><i/>;<br/><a NAME="E13:66_3_1_2"/><i><font color="Green">E125</font></i>: 
 ex <font color="Olive">t</font> being  <a href="msaterm.html#NM3">c-Term</a> of <font color="Maroon">A</font>,<font color="Maroon">V</font> st <br/>( <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp;  <a href="msaterm.html#K7" target="_self">the_sort_of</a> <font color="Olive">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> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp;  <a href="msaterm.html#K7" target="_self">the_sort_of</a> <font color="Olive">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="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i>, <a class="txt" href="msaterm.html#E1:66"><i><font color="Green">E26</font></i></a></i>;<br/><a NAME="E14:66_3_1_2"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E15:66_3_1_2"/><b>then </b>
<font color="Maroon">vt</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msaterm.html#K7" target="_self">the_sort_of</a> <font color="Maroon">t</font></span>)</span>
 <b>by </b><i><a class="txt" href="msaterm.html#E6"><i><font color="Green">Lemma46</font></i></a>, </i>;<br/><b>hence </b><a NAME="E16:66_3_1_2"/>
<span class="p1">(<span class="default"><a href="trees_3.html#K16">roots</a> <font color="Maroon">q</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#NK3" target="_self">*</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> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , , , , <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/></div><b>end;</b></div>
<a NAME="E10:66_3_1"/><b>then </b>
 <a href="trees_3.html#K16">roots</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#NK3" target="_self">*</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>
 
<b>by </b><i><a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<a NAME="E11:66_3_1"/><b>then </b>
 <a href="trees_3.html#K16">roots</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="funct_1.html#K1">.</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="pboole.html#D19">PBOOLE:def 19</a></i>;<br/>
<a NAME="E12:66_3_1"/><b>then </b>
 <a href="trees_3.html#K16">roots</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font></span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_1.html#D6">MSUALG_1:def 6</a></i>;<br/>
<a NAME="E13:66_3_1"/><b>then </b>
 <a href="trees_3.html#K16">roots</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 
<b>by </b><i><a class="txt" href="msaterm.html#E3:66"><i><font color="Green">E51</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<a NAME="E14:66_3_1"/><b>then </b>
(  <a href="trees_3.html#K16">roots</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">A</font> &amp;  <a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">A</font>, <a href="msualg_1.html#K4">Result</a> <font color="Maroon">o</font>,<font color="Maroon">A</font> )
 
<b>by </b><i><a class="ref" href="msualg_1.html#D9">MSUALG_1:def 9</a></i>;<br/>
<b>hence </b><a NAME="E15:66_3_1"/>
<font color="Maroon">vt</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msaterm.html#K7" target="_self">the_sort_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msaterm.html#K2" target="_self">Sym</a> <font color="Maroon">o</font>,<span class="p4">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="pboole.html#K2">\/</a> <font color="Maroon">V</font></span>)</span></span>)</span> <a href="msaterm.html#K6" target="_self">-tree</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="msaterm.html#E7"><i><font color="Green">Lemma50</font></i></a>, <a class="txt" href="msaterm.html#E1:66_3"><i><font color="Green">E52</font></i></a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E4:66"/>
for <font color="Olive">t</font> being  <a href="msaterm.html#NM3">c-Term</a> of <font color="Maroon">A</font>,<font color="Maroon">V</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#S3" target="_self">MSATERM:sch 3</a>(<a class="ref" href="msaterm.html#T1" target="_self">Th1</a>, , );<br/></i>
<b>hence </b><a NAME="E5:66"/>
for <font color="Olive">vt</font> being <a href="finset_1.html#V1">finite</a> <a href="trees_2.html#NM4">DecoratedTree</a>  st <font color="Olive">vt</font> <a href="msaterm.html#R1" target="_self">is_an_evaluation_of</a> <font color="Maroon">t</font>,<font color="Maroon">f</font> holds <br/><font color="Olive">vt</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msaterm.html#K7" target="_self">the_sort_of</a> <font color="Maroon">t</font></span>)</span>
 ;<br/>


</div>
