<?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">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">e1</font>, <font color="Olive">e2</font> being <a href="finset_1.html#V1">finite</a> <a href="trees_2.html#NM4">DecoratedTree</a>  st <font color="Olive">e1</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> &amp; <font color="Olive">e2</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">e1</font> <a href="hidden.html#R1">=</a> <font color="Olive">e2</font>;<br/>
<div><i><font color="Green">E26</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="65_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="E1:65_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="65_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">e1</font> be  <a href="finset_1.html#V1">finite</a> <a href="trees_2.html#NM4">DecoratedTree</a>, <font color="Maroon">e2</font> be  <a href="finset_1.html#V1">finite</a> <a href="trees_2.html#NM4">DecoratedTree</a>;<br/>

<b>assume </b><a NAME="E1:65_1_1"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">e1</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> &amp; <font color="Maroon">e2</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="E2:65_1_1"/><b>then </b><i><font color="Green">E28</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">e1</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</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> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">e2</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</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> &amp; <font color="Maroon">x</font> <a href="msaterm.html#K4" target="_self">-term</a> <font color="Maroon">A</font>,<font color="Maroon">V</font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="trees_4.html#K1">root-tree</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="trees_1.html#K2">elementary_tree</a> 0 &amp; <span class="p1">(<span class="default"><a href="trees_4.html#K1">root-tree</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span></span>)</span> <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">x</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span> )
 
<b>by </b><i>, <a class="ref" href="trees_4.html#T3">TREES_4:3</a></i>;<br/>
<a NAME="E3:65_1_1"/>
 <a href="xboole_0.html#K1">{}</a>  is   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">e1</font>
 
<b>by </b><i><a class="ref" href="trees_1.html#T47">TREES_1:47</a></i>;<br/>
<a NAME="E4:65_1_1"/><b>then </b>
( <font color="Maroon">e1</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">e2</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> )
 
<b>by </b><i><a class="ref" href="msaterm.html#T1" target="_self">Th1</a>, , </i>;<br/>
<a NAME="E5:65_1_1"/><b>then </b>
( <font color="Maroon">e1</font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K2">root-tree</a> <font color="Maroon">x</font> &amp; <font color="Maroon">e2</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="trees_4.html#T5">TREES_4:5</a></i>;<br/>
<b>hence </b><a NAME="E6:65_1_1"/>
<font color="Maroon">e1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font>
 ;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<div><i><font color="Green">E45</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="65_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="E1:65_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="65_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">e1</font> be  <a href="finset_1.html#V1">finite</a> <a href="trees_2.html#NM4">DecoratedTree</a>, <font color="Maroon">e2</font> be  <a href="finset_1.html#V1">finite</a> <a href="trees_2.html#NM4">DecoratedTree</a>;<br/>

<b>assume </b><a NAME="E1:65_2_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">e1</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> &amp; <font color="Maroon">e2</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="E2:65_2_1"/><b>then </b><i><font color="Green">E52</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">e1</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</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> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">e2</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</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> &amp; <font color="Maroon">v</font> <a href="msaterm.html#K5" target="_self">-term</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="trees_4.html#K1">root-tree</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="trees_1.html#K2">elementary_tree</a> 0 &amp; <span class="p1">(<span class="default"><a href="trees_4.html#K1">root-tree</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span></span>)</span> <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">v</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span> )
 
<b>by </b><i>, <a class="ref" href="trees_4.html#T3">TREES_4:3</a></i>;<br/>
<a NAME="E3:65_2_1"/>
 <a href="xboole_0.html#K1">{}</a>  is   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">e1</font>
 
<b>by </b><i><a class="ref" href="trees_1.html#T47">TREES_1:47</a></i>;<br/>
<a NAME="E4:65_2_1"/><b>then </b>
( <font color="Maroon">e1</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">e2</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> )
 
<b>by </b><i>, <a class="txt" href="msaterm.html#E2:65"><i><font color="Green">E45</font></i></a>, </i>;<br/>
<a NAME="E5:65_2_1"/><b>then </b>
( <font color="Maroon">e1</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> &amp; <font color="Maroon">e2</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#E2:65"><i><font color="Green">E45</font></i></a>, <a class="ref" href="trees_4.html#T5">TREES_4:5</a></i>;<br/>
<b>hence </b><a NAME="E6:65_2_1"/>
<font color="Maroon">e1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font>
 ;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<div><i><font color="Green">E53</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="65_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="E1:65_3"/><i><font color="Green">E54</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="E2:65_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="65_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e1</font> be  <a href="finset_1.html#V1">finite</a> <a href="trees_2.html#NM4">DecoratedTree</a>, <font color="Maroon">e2</font> be  <a href="finset_1.html#V1">finite</a> <a href="trees_2.html#NM4">DecoratedTree</a>;<br/>

<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>assume </b><a NAME="E1:65_3_1"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">e1</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> &amp; <font color="Maroon">e2</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">q1</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="E3:65_3_1"/><i><font color="Green">E63</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> &amp; <font color="Maroon">e1</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">q1</font></span>)</span></span>)</span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">q1</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">q1</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/>
<b>consider </b><font color="Maroon">q2</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="E5:65_3_1"/><i><font color="Green">E86</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> &amp; <font color="Maroon">e2</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">q2</font></span>)</span></span>)</span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">q2</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">q2</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><i><font color="Green">E87</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="65_3_1_1"><b>now </b></a><div class="add"><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="E1:65_3_1_1"/><i><font color="Green">E116</font></i>: 
<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">p</font>
 ;<br/><b>then 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/>;<br/><a NAME="E3:65_3_1_1"/><i><font color="Green">E118</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><a class="txt" href="msaterm.html#E7"><i><font color="Green">Lemma50</font></i></a>, </i>;<br/><b>then consider </b><font color="Maroon">vt1</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="E5:65_3_1_1"/><i><font color="Green">E119</font></i>: 
( <font color="Maroon">vt1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</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">vt1</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/><b>consider </b><font color="Maroon">vt2</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="E7:65_3_1_1"/><i><font color="Green">E120</font></i>: 
( <font color="Maroon">vt2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</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">vt2</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/><a NAME="E8:65_3_1_1"/>
<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><a class="txt" href="msaterm.html#E7"><i><font color="Green">Lemma50</font></i></a>, </i>;<br/><b>hence </b><a NAME="E9:65_3_1_1"/>
<font color="Maroon">q1</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> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</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>
 <b>by </b><i><a class="txt" href="msaterm.html#E2:65"><i><font color="Green">E45</font></i></a>, , <a class="txt" href="msaterm.html#E3:65"><i><font color="Green">E53</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:65_3_1"/><i><font color="Green">E121</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q1</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">q1</font></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q2</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">q2</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="65_3_1_2"><b>now </b></a><div class="add"><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="E1:65_3_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q1</font>
 ;<br/><a NAME="E2:65_3_1_2"/><b>then </b>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q1</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:65_3_1_2"/>
<font color="Maroon">q1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<a NAME="E9:65_3_1"/><b>then </b>
<font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font>
 
<b>by </b><i>, , <a class="txt" href="msaterm.html#E7"><i><font color="Green">Lemma50</font></i></a>, <a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a></i>;<br/>
<b>hence </b><a NAME="E10:65_3_1"/>
<font color="Maroon">e1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E4:65"/>
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="E5:65"/>
for <font color="Olive">e1</font>, <font color="Olive">e2</font> being <a href="finset_1.html#V1">finite</a> <a href="trees_2.html#NM4">DecoratedTree</a>  st <font color="Olive">e1</font> <a href="msaterm.html#R1" target="_self">is_an_evaluation_of</a> <font color="Maroon">t</font>,<font color="Maroon">f</font> &amp; <font color="Olive">e2</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">e1</font> <a href="hidden.html#R1">=</a> <font color="Olive">e2</font>
 ;<br/>


</div>
