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<b>let </b><font color="Maroon" title="c1">S</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> non <a href="msualg_1.html#V2" title="MSUALG_1:attr.2">void</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon" title="c2">A</font> be  <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">S</font>;<br/><b>let </b><font color="Maroon" title="c3">V</font> be    <a href="msaterm.html#M4" title="MSATERM:mode.4">Variables</a> of <font color="Maroon" title="c2">A</font>;<br/><b>let </b><font color="Maroon" title="c4">f</font> be    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <font color="Maroon" title="c3">V</font>,the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font>;<br/><b>let </b><font color="Maroon" title="c5">o</font> be   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font>;<br/><b>let </b><font color="Maroon" title="c6">p</font> be   <a href="msaterm.html#NM4" title="MSATERM:NM.4">ArgumentSeq</a> of <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">V</font>;<br/>











<a NAME="E1:63"/><i><font color="Green" title="E39">A1</font></i>: 
 <a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> <font color="Maroon" title="c5">o</font>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c3">V</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="ref" href="msafree.html#D11" title="MSAFREE:def.11">MSAFREE:def 11</a></i>;<br/>
<b>let </b><font color="Maroon" title="c7">vt</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="trees_2.html#NM4" title="TREES_2:NM.4">DecoratedTree</a>;<br/>

<b>assume </b><a NAME="E2:63"/><i><font color="Green" title="E40">A2</font></i>: 
<font color="Maroon" title="c7">vt</font> <a href="msaterm.html#R1" title="MSATERM:pred.1">is_an_evaluation_of</a> <span class="p1">(<span class="default"><a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> <font color="Maroon" title="c5">o</font>,<span class="p2">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c3">V</font></span>)</span></span>)</span> <a href="msaterm.html#K6" title="MSATERM:func.6">-tree</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c4">f</font>
 ;<br/>

<a NAME="E3:63"/><b>then </b><i><font color="Green" title="E41">A3</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">vt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c6">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="msaterm.html#E1:63"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="msaterm.html#D9" target="_self" title="MSATERM:def.9">Def9</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c9">q</font> being  <a href="trees_3.html#V6" title="TREES_3:attr.6">DTree-yielding</a> <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E5:63"/><i><font color="Green" title="E42">A4</font></i>: 
<font color="Maroon" title="c7">vt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x</font> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c9">q</font>
 <b>by </b><i><a class="ref" href="trees_9.html#T8" title="TREES_9:th.8">TREES_9:8</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c9">q</font>
;<br/>

<a NAME="E6:63"/>
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">vt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_3.html#K13" title="TREES_3:func.13">tree</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c9">q</font></span>)</span> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">vt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_3.html#K13" title="TREES_3:func.13">tree</a> <span class="p1">(<span class="default"><a href="trees_4.html#K11" title="TREES_4:func.11">doms</a> <font color="Maroon" title="c6">p</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="msaterm.html#E3:63"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="msaterm.html#E5:63"><i><font color="Green" title="E42">A4</font></i></a>, <a class="ref" href="trees_4.html#T10" title="TREES_4:th.10">TREES_4:10</a></i>;<br/>
<a NAME="E7:63"/><b>then </b><i><font color="Green" title="E43">A5</font></i>: 
(  <a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c9">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K11" title="TREES_4:func.11">doms</a> <font color="Maroon" title="c6">p</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c9">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">q</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="trees_4.html#K11" title="TREES_4:func.11">doms</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p</font> )
 
<b>by </b><i><a class="ref" href="trees_3.html#T40" title="TREES_3:th.40">TREES_3:40</a>, <a class="ref" href="trees_3.html#T53" title="TREES_3:th.53">TREES_3:53</a></i>;<br/>
<b>hence </b><a NAME="E8:63"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c10">r</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">vt</font> <b>by </b><i><a class="ref" href="trees_1.html#T47" title="TREES_1:th.47">TREES_1:47</a></i>;<br/>
<a NAME="E10:63"/>
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="ref" href="trees_4.html#D4" title="TREES_4:def.4">TREES_4:def 4</a></i>;<br/>
<b>then </b><font color="Maroon" title="c7">vt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">r</font> = 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="trees_9.html#K2" title="TREES_9:func.2">succ</a> <font color="Maroon" title="c7">vt</font>,<font color="Maroon" title="c10">r</font></span>)</span>
<b>by </b><i><a class="txt" href="msaterm.html#E1:63"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="msaterm.html#E2:63"><i><font color="Green" title="E40">A2</font></i></a>, <a class="ref" href="msaterm.html#D9" target="_self" title="MSATERM:def.9">Def9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <font color="Maroon" title="c9">q</font></span>)</span>
<b>by </b><i><a class="txt" href="msaterm.html#E5:63"><i><font color="Green" title="E42">A4</font></i></a>, <a class="ref" href="trees_9.html#T31" title="TREES_9:th.31">TREES_9:31</a></i>
;<br/>
<b>hence </b><a NAME="E12:63"/>
<font color="Maroon" title="c7">vt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <font color="Maroon" title="c9">q</font></span>)</span></span>)</span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c9">q</font>
 <b>by </b><i><a class="txt" href="msaterm.html#E5:63"><i><font color="Green" title="E42">A4</font></i></a>, <a class="ref" href="trees_4.html#D4" title="TREES_4:def.4">TREES_4:def 4</a></i>;<br/>

<b>let </b><font color="Maroon" title="c11">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>let </b><font color="Maroon" title="c12">t</font> be   <a href="msaterm.html#NM3" title="MSATERM:NM.3">c-Term</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">V</font>;<br/>



<b>assume </b><a NAME="E13:63"/><i><font color="Green" title="E44">A6</font></i>: 
( <font color="Maroon" title="c11">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">p</font> &amp; <font color="Maroon" title="c12">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c13">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E15:63"/><i><font color="Green" title="E45">A7</font></i>: 
( <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c13">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p</font> )
 <b>by </b><i><a class="txt" href="msaterm.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">u</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c12">t</font> <b>by </b><i><a class="ref" href="trees_1.html#T47" title="TREES_1:th.47">TREES_1:47</a></i>;<br/>
<a NAME="E17:63"/>
<span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c13">k</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c13">k</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">FINSEQ_1:47</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c15">r</font> = <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c13">k</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> as   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c7">vt</font> <b>by </b><i><a class="txt" href="msaterm.html#E3:63"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="msaterm.html#E13:63"><i><font color="Green" title="E44">A6</font></i></a>, <a class="txt" href="msaterm.html#E15:63"><i><font color="Green" title="E45">A7</font></i></a>, <a class="ref" href="trees_4.html#T11" title="TREES_4:th.11">TREES_4:11</a></i>;<br/>
<b>take </b><font color="Maroon" title="c16">e</font> = <font color="Maroon" title="c7">vt</font> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <font color="Maroon" title="c15">r</font>;<br/>

<b>thus </b><a NAME="E19:63"/>
<font color="Maroon" title="c16">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font>
 <b>by </b><i><a class="txt" href="msaterm.html#E5:63"><i><font color="Green" title="E42">A4</font></i></a>, <a class="txt" href="msaterm.html#E7:63"><i><font color="Green" title="E43">A5</font></i></a>, <a class="txt" href="msaterm.html#E15:63"><i><font color="Green" title="E45">A7</font></i></a>, <a class="ref" href="trees_4.html#D4" title="TREES_4:def.4">TREES_4:def 4</a></i>;<br/>

<b>reconsider </b><font color="Maroon" title="c17">r1</font> = <font color="Maroon" title="c15">r</font> as   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c6">p</font></span>)</span> <b>by </b><i><a class="txt" href="msaterm.html#E1:63"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="msaterm.html#E2:63"><i><font color="Green" title="E40">A2</font></i></a>, <a class="ref" href="msaterm.html#D9" target="_self" title="MSATERM:def.9">Def9</a></i>;<br/>
<a NAME="E21:63"/>
<font color="Maroon" title="c12">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <font color="Maroon" title="c17">r1</font>
 
<b>by </b><i><a class="txt" href="msaterm.html#E13:63"><i><font color="Green" title="E44">A6</font></i></a>, <a class="txt" href="msaterm.html#E15:63"><i><font color="Green" title="E45">A7</font></i></a>, <a class="ref" href="trees_4.html#D4" title="TREES_4:def.4">TREES_4:def 4</a></i>;<br/>
<b>hence </b><a NAME="E22:63"/>
<font color="Maroon" title="c16">e</font> <a href="msaterm.html#R1" title="MSATERM:pred.1">is_an_evaluation_of</a> <font color="Maroon" title="c12">t</font>,<font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="txt" href="msaterm.html#E1:63"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="msaterm.html#E2:63"><i><font color="Green" title="E40">A2</font></i></a>, <a class="ref" href="msaterm.html#T34" target="_self" title="MSATERM:th.34">Th34</a></i>;<br/>


</div>
