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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">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>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>let </b><font color="Maroon">q</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/>













<b>assume </b><a NAME="E1:72"/><i><font color="Green">E26</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> &amp; ( for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  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> holds <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">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/><font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">t</font> <a href="msaterm.html#K9" target="_self">@</a> <font color="Maroon">f</font> ) )
 ;<br/>

<b>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="E3:72"/><i><font color="Green">E27</font></i>: 
<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>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">r</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="E5:72"/><i><font color="Green">E28</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> &amp; <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">r</font></span>)</span></span>)</span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">r</font> &amp; ( 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">r</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><a class="txt" href="msaterm.html#E6"><i><font color="Green">Lemma46</font></i></a>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:72_1"/><i><font color="Green">E45</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> )
 <b>by </b><i><a class="txt" href="msaterm.html#E1:72_1"><i><font color="Green">E45</font></i></a>, , <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E2:72_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">r</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#NM3">c-Term</a> of <font color="Maroon">A</font>,<font color="Maroon">V</font> <b>by </b><i>, <a class="txt" href="msaterm.html#E3:72"><i><font color="Green">E27</font></i></a></i>;<br/><b>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="E5:72_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">vt</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &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>, , </i>;<br/><b>reconsider </b><font color="Maroon">T</font> = <font color="Maroon">vt</font> as   <a href="trees_2.html#NM4">DecoratedTree</a> ;<br/><b>take </b><font color="Maroon">T</font> = <font color="Maroon">T</font>;<br/><b>thus </b><a NAME="E7:72_1"/>
<font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="msaterm.html#E7"><i><font color="Green">Lemma50</font></i></a></i>;<br/><b>thus </b><font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<font color="Maroon">t</font> <a href="msaterm.html#K9" target="_self">@</a> <font color="Maroon">f</font>
<b>by </b><i><a class="txt" href="msaterm.html#E1:72_1"><i><font color="Green">E45</font></i></a>, , </i>
<br/>.= 
<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#E7"><i><font color="Green">Lemma50</font></i></a>, </i>
;<br/></div><b>end;</b></div>
<a NAME="E7:72"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="trees_3.html#K16">roots</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="trees_3.html#D18">TREES_3:def 18</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msaterm.html#K2" target="_self">Sym</a> <font color="Maroon">o</font>,<span class="p3">(<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> <a href="msaterm.html#K9" target="_self">@</a> <font color="Maroon">f</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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> <font color="Maroon">q</font></span>)</span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">r</font></span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="txt" href="msaterm.html#E6"><i><font color="Green">Lemma46</font></i></a>, , </i>
<br/>.= 
<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> <font color="Maroon">q</font>
<b>by </b><i><a class="ref" href="trees_4.html#D4">TREES_4:def 4</a></i>
;<br/><br/>


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