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

<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> be  <a href="relat_1.html#V2">V2</a>  <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>







<b>assume </b><a NAME="E1:5"/><i><font color="Green">E6</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E2:5"/><i><font color="Green">E7</font></i>: 
 <a href="msafree.html#K7">FreeSort</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> <a href="msaterm.html#K1">-Terms</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="msaterm.html#T12">MSATERM:12</a></i>;<br/>
<a NAME="E3:5"/>
 <a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#G3">MSAlgebra</a>(# <span class="p1">(<span class="default"><a href="msafree.html#K8">FreeSort</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msafree.html#K10">FreeOper</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="msafree.html#D16">MSAFREE:def 16</a></i>;<br/>
<a NAME="E4:5"/><b>then </b><i><font color="Green">E8</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="msafree.html#K7">FreeSort</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="msafree.html#D13">MSAFREE:def 13</a></i>;<br/>
<b>thus </b><a NAME="E5:5"/>
( ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> st  <a href="msafree2.html#K9">depth</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> implies ex <font color="Olive">b<sub>1</sub></font> being <a href="graph_1.html#V8">directed</a>  <a href="graph_1.html#M1">Chain</a> of  <a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">c<sub>1</sub></font> st <br/>(  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">c<sub>6</sub></font> being    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font><b> such that </b><a NAME="E1:5_1"/><i><font color="Green">E9</font></i>: 
 <a href="msafree2.html#K9">depth</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:5_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree.html#K7">FreeSort</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="msscyc_2.html#E4:5"><i><font color="Green">E8</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="msscyc_2.html#E2:5"><i><font color="Green">E7</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being  <a href="finset_1.html#V1">finite</a> <a href="trees_2.html#NM4">DecoratedTree</a>, <font color="Maroon">c<sub>9</sub></font> being  <a href="finset_1.html#V1">finite</a> <a href="trees_1.html#NM1">Tree</a><b> such that </b><br/><a NAME="E5:5_1"/><i><font color="Green">E10</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>8</sub></font> &amp;  <a href="msafree2.html#K9">depth</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="trees_1.html#K6">height</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="ref" href="msafree2.html#D14">MSAFREE2:def 14</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:5_1"/><i><font color="Green">E11</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="trees_1.html#K6">height</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="ref" href="trees_1.html#D15">TREES_1:def 15</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>11</sub></font> = the <a href="graph_1.html#U2">Edges</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_1.html#K13">*</a> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="trees_2.html#K5">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="trees_2.html#K5">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; ex <font color="Olive">b<sub>3</sub></font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>4</sub></font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>5</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U2">Edges</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_1.html#K13">*</a>  st <br/>( <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Olive">b<sub>5</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>3</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="msaterm.html#K7">the_sort_of</a> <font color="Olive">b<sub>2</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> ) );<br/>
<a NAME="E8:5_1"/><i><font color="Green">E12</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> holds <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U2">Edges</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_1.html#K13">*</a> ex <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U2">Edges</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_1.html#K13">*</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:5_1_1"/><i><font color="Green">E13</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<a NAME="E2:5_1_1"/><b>then </b><i><font color="Green">E14</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>13</sub></font> be    <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U2">Edges</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_1.html#K13">*</a> ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">c<sub>6</sub></font> <b>by </b><i><a class="txt" href="msscyc_2.html#E5:5_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="msscyc_2.html#E7:5_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="msscyc_1.html#T19">MSSCYC_1:19</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="trees_2.html#K5">|</a> <font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="trees_2.html#K5">|</a> <font color="Maroon">c<sub>15</sub></font> as   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="msaterm.html#T29">MSATERM:29</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>12</sub></font>;<br/>
<a NAME="E5:5_1_1"/><i><font color="Green">E15</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="msscyc_2.html#E1:5_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="msscyc_2.html#E5:5_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="msscyc_2.html#E7:5_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="msscyc_2.html#E1:5_1_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>;<br/>
<a NAME="E6:5_1_1"/><i><font color="Green">E16</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="msscyc_2.html#E2:5_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E7:5_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="msscyc_2.html#E2:5_1_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="msscyc_2.html#E5:5_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E8:5_1_1"/><b>then </b><i><font color="Green">E17</font></i>: 
not <font color="Maroon">c<sub>10</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>12</sub></font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E9:5_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="rfinseq.html#T21">RFINSEQ:21</a></i>;<br/>
<a NAME="E10:5_1_1"/><b>then </b><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> <a href="trees_1.html#K4">|</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="msscyc_2.html#E5:5_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="msscyc_2.html#E7:5_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="trees_1.html#D9">TREES_1:def 9</a></i>;<br/>
<a NAME="E11:5_1_1"/><b>then </b><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>16</sub></font>
 
<b>by </b><i><a class="txt" href="msscyc_2.html#E5:5_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="trees_2.html#D11">TREES_2:def 11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:5_1_1_1"/>
<font color="Maroon">c<sub>16</sub></font> is <a href="trees_9.html#V1">root</a>
 ;<br/><a NAME="E2:5_1_1_1"/><b>then </b><i><font color="Green">E20</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a>  <a href="trees_1.html#K2">elementary_tree</a> 0
 <b>by </b><i><a class="ref" href="trees_9.html#D1">TREES_9:def 1</a></i>;<br/><a NAME="E3:5_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="trees_1.html#K6">height</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="msscyc_2.html#E11:5_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="trees_1.html#D15">TREES_1:def 15</a></i>;<br/><b>hence </b><a NAME="E4:5_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="msscyc_2.html#E5:5_1_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="msscyc_2.html#E6:5_1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="msscyc_2.html#E2:5_1_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="trees_1.html#T79">TREES_1:79</a></i>;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>19</sub></font> being   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E14:5_1_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <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">c<sub>19</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="msscyc_1.html#T20">MSSCYC_1:20</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>20</sub></font> being    <a href="msaterm.html#M1">ArgumentSeq</a> of  <a href="msaterm.html#K2">Sym</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E16:5_1_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="msscyc_2.html#E14:5_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="msaterm.html#T10">MSATERM:10</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>21</sub></font> being   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">c<sub>22</sub></font> being   <a href="trees_2.html#NM4">DecoratedTree</a>, <font color="Maroon">c<sub>23</sub></font> being   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">c<sub>22</sub></font><b> such that </b><br/><a NAME="E18:5_1_1"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>21</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>23</sub></font> )
 <b>by </b><i><a class="txt" href="msscyc_2.html#E8:5_1_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="msscyc_2.html#E11:5_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="msscyc_2.html#E16:5_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="trees_4.html#T11">TREES_4:11</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>24</sub></font> =  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font>;<br/>
<a NAME="E19:5_1_1"/><i><font color="Green">E23</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a>  <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">c<sub>19</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="msaterm.html#T22">MSATERM:22</a></i>;<br/>
<a NAME="E20:5_1_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font> )
 
<b>by </b><i><a class="txt" href="msscyc_2.html#E18:5_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E21:5_1_1"/><b>then </b><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <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">c<sub>19</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="msscyc_2.html#E19:5_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>25</sub></font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="dtconstr.html#T2">DTCONSTR:2</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>26</sub></font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>25</sub></font></span><a href="tarski.html#K4">]</a></span>;<br/>
<a NAME="E23:5_1_1"/><i><font color="Green">E25</font></i>: 
the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="ref" href="msualg_1.html#D6">MSUALG_1:def 6</a></i>;<br/>
<a NAME="E24:5_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>25</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="msscyc_2.html#K1" target="_self">InducedEdges</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="msscyc_2.html#E21:5_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="msscyc_2.html#D1" target="_self">Def1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>27</sub></font> = the <a href="graph_1.html#U2">Edges</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>28</sub></font> = <font color="Maroon">c<sub>13</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>27</sub></font> <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>29</sub></font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>25</sub></font></span><a href="tarski.html#K4">]</a></span> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>27</sub></font> <b>by </b><i><a class="txt" href="msscyc_2.html#E21:5_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="msscyc_2.html#E23:5_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="msscyc_2.html#D1" target="_self">Def1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>30</sub></font> = <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>29</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>28</sub></font> as    <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U2">Edges</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_1.html#K13">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>30</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>16</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>17</sub></font>
;<br/>

<b>thus </b><a NAME="E29:5_1_1"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="trees_2.html#K5">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="trees_2.html#K5">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 ;<br/>

<b>take </b>
<font color="Maroon">c<sub>19</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>25</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>13</sub></font>
;<br/>

<b>thus </b><a NAME="E30:5_1_1"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>25</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>13</sub></font> )
 ;<br/>

<b>thus </b><a NAME="E31:5_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="msscyc_2.html#E14:5_1_1"><i><font color="Green">E20</font></i></a></i>;<br/>

<a NAME="E32:5_1_1"/><i><font color="Green">E26</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="msscyc_2.html#E8:5_1_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/>
<i><font color="Green">E27</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> 1 = 
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
<b>by </b><i><a class="txt" href="msscyc_2.html#E8:5_1_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="finseq_5.html#T23">FINSEQ_5:23</a></i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>
<b>by </b><i><a class="txt" href="msscyc_2.html#E32:5_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/>
<b>reconsider </b><font color="Maroon">c<sub>31</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>12</sub></font> as   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">c<sub>16</sub></font> <b>by </b><i><a class="txt" href="msscyc_2.html#E5:5_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="msscyc_2.html#E10:5_1_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="trees_2.html#D11">TREES_2:def 11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>32</sub></font> = <font color="Maroon">c<sub>31</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">c<sub>16</sub></font> <b>by </b><i><a class="ref" href="msscyc_1.html#T19">MSSCYC_1:19</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>33</sub></font> = <font color="Maroon">c<sub>16</sub></font> <a href="msaterm.html#K8">|</a> <font color="Maroon">c<sub>32</sub></font> as   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> ;<br/>
<a NAME="E37:5_1_1"/>
<font color="Maroon">c<sub>31</sub></font> <a href="finseq_1.html#K17">|</a> 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>21</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="msscyc_2.html#E18:5_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="msscyc_2.html#E33:5_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="finseq_1.html#T58">FINSEQ_1:58</a></i>;<br/>
<a NAME="E38:5_1_1"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="msscyc_2.html#E16:5_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="msscyc_2.html#E18:5_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="trees_4.html#D4">TREES_4:def 4</a></i>;<br/>
<a NAME="E39:5_1_1"/><i><font color="Green">E29</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="msscyc_2.html#E1:5_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="msscyc_2.html#E5:5_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="msscyc_2.html#E7:5_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="msscyc_2.html#E2:5_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E40:5_1_1"/><i><font color="Green">E30</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E41:5_1_1"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>12</sub></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">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="msscyc_2.html#E39:5_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E42:5_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>12</sub></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">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="msscyc_2.html#E1:5_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="msscyc_2.html#E5:5_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="msscyc_2.html#E7:5_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="msscyc_2.html#E2:5_1_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="msscyc_2.html#E40:5_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E43:5_1_1"/><i><font color="Green">E33</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="msscyc_2.html#E2:5_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/>
<i><font color="Green">E34</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>12</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="msscyc_2.html#E43:5_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_1.html#T82">FUNCT_1:82</a></i>
<br/>.= 
<font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>
;<br/>
<font color="Maroon">c<sub>32</sub></font> = 
<span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>
<b>by </b><i><a class="txt" href="msscyc_2.html#E1:5_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="msscyc_2.html#E5:5_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="msscyc_2.html#E7:5_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="msscyc_2.html#E1:5_1_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="msscyc_2.html#E32:5_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="msscyc_2.html#E33:5_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
<b>by </b><i><a class="txt" href="msscyc_2.html#E42:5_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
<b>by </b><i><a class="txt" href="msscyc_2.html#E41:5_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="finseq_4.html#T85">FINSEQ_4:85</a></i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>
<b>by </b><i><a class="txt" href="msscyc_2.html#E41:5_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/>
<a NAME="E46:5_1_1"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>32</sub></font>
 
<b>by </b><i><a class="txt" href="msscyc_2.html#E39:5_1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="msscyc_2.html#E44:5_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="rfinseq.html#T20">RFINSEQ:20</a></i>;<br/>
<a NAME="E47:5_1_1"/><b>then </b>
<font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="ref" href="trees_9.html#T3">TREES_9:3</a></i>;<br/>
<b>hence </b><a NAME="E48:5_1_1"/>
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a>  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="msscyc_2.html#E19:5_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="msscyc_2.html#E21:5_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="msscyc_2.html#E38:5_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="msaterm.html#T23">MSATERM:23</a></i>;<br/>

<b>thus </b><a NAME="E49:5_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>25</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="msscyc_2.html#E21:5_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="msscyc_2.html#E23:5_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="msscyc_2.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:5_1"/>
not <font color="Maroon">c<sub>6</sub></font> is <a href="trees_9.html#V1">root</a>
 
<b>by </b><i><a class="txt" href="msscyc_2.html#E1:5"><i><font color="Green">E6</font></i></a>, <a class="txt" href="msscyc_2.html#E1:5_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="msscyc_2.html#E5:5_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="trees_1.html#T79">TREES_1:79</a>, <a class="ref" href="trees_9.html#D1">TREES_9:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E11:5_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <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">c<sub>12</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="msscyc_1.html#T20">MSSCYC_1:20</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="msaterm.html#M1">ArgumentSeq</a> of  <a href="msaterm.html#K2">Sym</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E13:5_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="msscyc_2.html#E11:5_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="msaterm.html#T10">MSATERM:10</a></i>;<br/>
<a NAME="E14:5_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a>, <a class="txt" href="msscyc_2.html#E1:5"><i><font color="Green">E6</font></i></a>, <a class="txt" href="msscyc_2.html#E1:5_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="msscyc_2.html#E5:5_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="msscyc_2.html#E7:5_1"><i><font color="Green">E11</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">c<sub>15</sub></font> being   <a href="trees_2.html#NM4">DecoratedTree</a>, <font color="Maroon">c<sub>16</sub></font> being   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">c<sub>15</sub></font><b> such that </b><br/><a NAME="E16:5_1"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="msscyc_2.html#E5:5_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="msscyc_2.html#E7:5_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="msscyc_2.html#E13:5_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="trees_4.html#T11">TREES_4:11</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>17</sub></font> =  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>12</sub></font>;<br/>
<a NAME="E17:5_1"/><i><font color="Green">E17</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <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">c<sub>12</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="msaterm.html#T22">MSATERM:22</a></i>;<br/>
<a NAME="E18:5_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font> )
 
<b>by </b><i><a class="txt" href="msscyc_2.html#E16:5_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E19:5_1"/><b>then </b><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <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">c<sub>12</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="msscyc_2.html#E17:5_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>18</sub></font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="dtconstr.html#T2">DTCONSTR:2</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>19</sub></font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="tarski.html#K4">]</a></span>;<br/>
<a NAME="E21:5_1"/><i><font color="Green">E19</font></i>: 
the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="ref" href="msualg_1.html#D6">MSUALG_1:def 6</a></i>;<br/>
<a NAME="E22:5_1"/><b>then </b><i><font color="Green">E20</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="msscyc_2.html#K1" target="_self">InducedEdges</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="msscyc_2.html#E19:5_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="msscyc_2.html#D1" target="_self">Def1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>20</sub></font> = the <a href="graph_1.html#U2">Edges</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>21</sub></font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="tarski.html#K4">]</a></span> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>20</sub></font> <b>by </b><i><a class="txt" href="msscyc_2.html#E19:5_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="msscyc_2.html#E21:5_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="msscyc_2.html#D1" target="_self">Def1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>22</sub></font> = <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>21</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> as    <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U2">Edges</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_1.html#K13">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>23</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U2">Edges</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_1.html#K13">*</a> <b> such that </b><br/><a NAME="E27:5_1"/><i><font color="Green">E21</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; ( <font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>] ) )
 <b>from </b><i><a class="ref" href="recdef_1.html#S6">RECDEF_1:sch 6</a>(<a class="txt" href="msscyc_2.html#E8:5_1"><i><font color="Green">E12</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> implies ex <font color="Olive">b<sub>1</sub></font> being <a href="graph_1.html#V8">directed</a>  <a href="graph_1.html#M1">Chain</a> of  <a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>2</sub></font> being  <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="trees_2.html#K5">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="msaterm.html#K7">the_sort_of</a> <font color="Olive">b<sub>2</sub></font> ) );<br/>
<a NAME="E28:5_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
;<br/>
<a NAME="E29:5_1"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>24</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:5_1_2"/><i><font color="Green">E24</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>24</sub></font> &amp; <font color="Maroon">c<sub>24</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> implies ex <font color="Olive">b<sub>1</sub></font> being <a href="graph_1.html#V8">directed</a>  <a href="graph_1.html#M1">Chain</a> of  <a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>2</sub></font> being  <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="trees_2.html#K5">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="msaterm.html#K7">the_sort_of</a> <font color="Olive">b<sub>2</sub></font> ) )
 ;<br/>

<b>assume </b><a NAME="E2:5_1_2"/><i><font color="Green">E25</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<a NAME="E3:5_1_2"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E4:5_1_2"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="msscyc_2.html#E2:5_1_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:5_1_2_1"/>
( <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_2_1_1"><b>suppose </b></a><a NAME="E1:5_1_2_1_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">c<sub>25</sub></font> = <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="finseq_1.html#K9">*&gt;</a></span> as  <a href="graph_1.html#V8">directed</a>  <a href="graph_1.html#M1">Chain</a> of  <a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="msscyc_2.html#E22:5_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="msscyc_1.html#T5">MSSCYC_1:5</a></i>;<br/><a NAME="E3:5_1_2_1_1"/><i><font color="Green">E29</font></i>: 
 <a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p2">(<span class="default">the <a href="graph_1.html#U3">Source</a> of <span class="p3">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<span class="p2">(<span class="default">the <a href="graph_1.html#U4">Target</a> of <span class="p3">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="msscyc_2.html#E22:5_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="msscyc_1.html#T7">MSSCYC_1:7</a></i>;<br/><i><font color="Green">E30</font></i>: <span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>
<br/>.= 
the <a href="graph_1.html#U4">Target</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="msscyc_2.html#E3:5_1_2_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
<br/>.= 
the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> </span>)</span>
<b>by </b><i><a class="txt" href="msscyc_2.html#E22:5_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="msscyc_2.html#D3" target="_self">Def3</a></i>
<br/>.= 
the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
 <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="ref" href="msualg_1.html#D7">MSUALG_1:def 7</a></i>
<br/>.= 
 <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="msscyc_2.html#E11:5_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="msaterm.html#T17">MSATERM:17</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="msscyc_2.html#E4:5"><i><font color="Green">E8</font></i></a>, <a class="ref" href="msaterm.html#D5">MSATERM:def 5</a></i>
;<br/><b>reconsider </b><font color="Maroon">c<sub>26</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">c<sub>6</sub></font> <b>by </b><i><a class="txt" href="msscyc_2.html#E5:5_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="msscyc_2.html#E7:5_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="msscyc_1.html#T19">MSSCYC_1:19</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>27</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="trees_2.html#K5">|</a> <font color="Maroon">c<sub>26</sub></font> as   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="msaterm.html#T29">MSATERM:29</a></i>;<br/><a NAME="E7:5_1_2_1_1"/><i><font color="Green">E31</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="msscyc_2.html#E14:5_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>28</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K5">NAT</a>  ;<br/><font color="Maroon">c<sub>10</sub></font> <a href="finseq_1.html#K17">|</a> 1 = 
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
<b>by </b><i><a class="txt" href="msscyc_2.html#E14:5_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="finseq_5.html#T23">FINSEQ_5:23</a></i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>
<b>by </b><i><a class="txt" href="msscyc_2.html#E7:5_1_2_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span>
<b>by </b><i><a class="txt" href="msscyc_2.html#E16:5_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="finseq_1.html#T58">FINSEQ_1:58</a></i>
;<br/><a NAME="E10:5_1_2_1_1"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="msscyc_2.html#E13:5_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="msscyc_2.html#E16:5_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="trees_4.html#D4">TREES_4:def 4</a></i>;<br/><i><font color="Green">E33</font></i>: <span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 1 = 
the <a href="graph_1.html#U3">Source</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="msscyc_2.html#E3:5_1_2_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="txt" href="msscyc_2.html#E22:5_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="msscyc_2.html#D2" target="_self">Def2</a></i>
<br/>.= 
<font color="Maroon">c<sub>18</sub></font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
 <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>27</sub></font>
<b>by </b><i><a class="txt" href="msscyc_2.html#E17:5_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="msscyc_2.html#E19:5_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="msscyc_2.html#E10:5_1_2_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="msaterm.html#T23">MSATERM:23</a></i>
;<br/><b>take </b>
<font color="Maroon">c<sub>25</sub></font>
;<br/><b>take </b>
<font color="Maroon">c<sub>27</sub></font>
;<br/><b>thus </b><a NAME="E12:5_1_2_1_1"/>
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="msscyc_2.html#E1:5"><i><font color="Green">E6</font></i></a>, <a class="txt" href="msscyc_2.html#E27:5_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="msscyc_2.html#E1:5_1_2_1_1"><i><font color="Green">E28</font></i></a></i>;<br/><b>thus </b><a NAME="E13:5_1_2_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="msscyc_2.html#E1:5_1_2_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/><b>thus </b><a NAME="E14:5_1_2_1_1"/>
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="trees_2.html#K5">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="msscyc_2.html#E1:5_1_2_1_1"><i><font color="Green">E28</font></i></a></i>;<br/><b>thus </b><a NAME="E15:5_1_2_1_1"/>
<span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="msscyc_2.html#E4:5_1_2_1_1"><i><font color="Green">E30</font></i></a></i>;<br/><b>thus </b><a NAME="E16:5_1_2_1_1"/>
<span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>27</sub></font>
 <b>by </b><i><a class="txt" href="msscyc_2.html#E11:5_1_2_1_1"><i><font color="Green">E33</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_2_1_2"><b>suppose </b></a><a NAME="E1:5_1_2_1_2"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:5_1_2_1_2"/><b>then </b><i><font color="Green">E29</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>25</sub></font> being  <a href="graph_1.html#V8">directed</a>  <a href="graph_1.html#M1">Chain</a> of  <a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>26</sub></font> being   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E4:5_1_2_1_2"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="trees_2.html#K5">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>26</sub></font> )
 <b>by </b><i><a class="txt" href="msscyc_2.html#E1:5_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="msscyc_2.html#E2:5_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="msscyc_2.html#E3:5_1_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="msscyc_2.html#E1:5_1_2_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>27</sub></font>, <font color="Maroon">c<sub>28</sub></font> being   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E6:5_1_2_1_2"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="trees_2.html#K5">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="trees_2.html#K5">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 <b>and </b><br/><a NAME="E7:5_1_2_1_2"/><i><font color="Green">E32</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>2</sub></font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U2">Edges</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_1.html#K13">*</a>  st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Olive">b<sub>3</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>28</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="msscyc_2.html#E27:5_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="msscyc_2.html#E4:5_1_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="msscyc_2.html#E2:5_1_2_1_2"><i><font color="Green">E29</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>29</sub></font> being   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>30</sub></font> being   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>31</sub></font> being    <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U2">Edges</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_1.html#K13">*</a> <b> such that </b><br/><a NAME="E9:5_1_2_1_2"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>29</sub></font>,<font color="Maroon">c<sub>30</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>31</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>29</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a>  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>28</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>29</sub></font>,<font color="Maroon">c<sub>30</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="msscyc_2.html#E7:5_1_2_1_2"><i><font color="Green">E32</font></i></a></i>;<br/><b>set </b><font color="Maroon">c<sub>32</sub></font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>29</sub></font>,<font color="Maroon">c<sub>30</sub></font></span><a href="tarski.html#K4">]</a></span>;<br/><b>reconsider </b><font color="Maroon">c<sub>33</sub></font> = <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>29</sub></font>,<font color="Maroon">c<sub>30</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="finseq_1.html#K9">*&gt;</a></span> as  <a href="graph_1.html#V8">directed</a>  <a href="graph_1.html#M1">Chain</a> of  <a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="msscyc_2.html#E9:5_1_2_1_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="msscyc_1.html#T5">MSSCYC_1:5</a></i>;<br/><a NAME="E11:5_1_2_1_2"/><i><font color="Green">E34</font></i>: 
not <font color="Maroon">c<sub>25</sub></font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="txt" href="msscyc_2.html#E1:5_1_2_1_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="msscyc_2.html#E4:5_1_2_1_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E12:5_1_2_1_2"/><i><font color="Green">E35</font></i>: 
not  <a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">c<sub>1</sub></font> is <a href="msscyc_1.html#V2">empty</a>
 <b>by </b><i><a class="txt" href="msscyc_2.html#E9:5_1_2_1_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="msscyc_1.html#D3">MSSCYC_1:def 3</a></i>;<br/><a NAME="E13:5_1_2_1_2"/><i><font color="Green">E36</font></i>: 
 <a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p2">(<span class="default">the <a href="graph_1.html#U3">Source</a> of <span class="p3">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>29</sub></font>,<font color="Maroon">c<sub>30</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<span class="p2">(<span class="default">the <a href="graph_1.html#U4">Target</a> of <span class="p3">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>29</sub></font>,<font color="Maroon">c<sub>30</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="msscyc_2.html#E9:5_1_2_1_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="msscyc_1.html#T7">MSSCYC_1:7</a></i>;<br/><i><font color="Green">E37</font></i>: <span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">c<sub>33</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>33</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">c<sub>33</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>
<br/>.= 
the <a href="graph_1.html#U4">Target</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>29</sub></font>,<font color="Maroon">c<sub>30</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="msscyc_2.html#E13:5_1_2_1_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
<br/>.= 
the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>29</sub></font>,<font color="Maroon">c<sub>30</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> </span>)</span>
<b>by </b><i><a class="txt" href="msscyc_2.html#E9:5_1_2_1_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="msscyc_2.html#D3" target="_self">Def3</a></i>
<br/>.= 
the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>29</sub></font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
 <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>29</sub></font>
<b>by </b><i><a class="ref" href="msualg_1.html#D7">MSUALG_1:def 7</a></i>
<br/>.= 
 <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>27</sub></font>
<b>by </b><i><a class="txt" href="msscyc_2.html#E9:5_1_2_1_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="msaterm.html#T17">MSATERM:17</a></i>
;<br/><b>then reconsider </b><font color="Maroon">c<sub>34</sub></font> = <font color="Maroon">c<sub>33</sub></font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>25</sub></font> as  <a href="graph_1.html#V8">directed</a>  <a href="graph_1.html#M1">Chain</a> of  <a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="msscyc_2.html#E4:5_1_2_1_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="msscyc_2.html#E6:5_1_2_1_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="msscyc_2.html#E11:5_1_2_1_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="msscyc_2.html#E12:5_1_2_1_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="msscyc_1.html#T15">MSSCYC_1:15</a></i>;<br/><a NAME="E16:5_1_2_1_2"/><i><font color="Green">E38</font></i>: 
not <font color="Maroon">c<sub>34</sub></font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="txt" href="msscyc_2.html#E4:5_1_2_1_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="msscyc_2.html#E6:5_1_2_1_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="msscyc_2.html#E11:5_1_2_1_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="msscyc_2.html#E12:5_1_2_1_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="msscyc_2.html#E14:5_1_2_1_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="msscyc_1.html#T15">MSSCYC_1:15</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>34</sub></font>
;<br/><b>take </b>
<font color="Maroon">c<sub>28</sub></font>
;<br/><b>thus </b><a NAME="E17:5_1_2_1_2"/>
<font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="msscyc_2.html#E4:5_1_2_1_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="msscyc_2.html#E9:5_1_2_1_2"><i><font color="Green">E33</font></i></a></i>;<br/><b>thus </b> <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>34</sub></font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>33</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>24</sub></font>
<b>by </b><i><a class="txt" href="msscyc_2.html#E4:5_1_2_1_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>
;<br/><b>thus </b><a NAME="E19:5_1_2_1_2"/>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="trees_2.html#K5">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="msscyc_2.html#E6:5_1_2_1_2"><i><font color="Green">E31</font></i></a></i>;<br/><b>thus </b><a NAME="E20:5_1_2_1_2"/>
<span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">c<sub>34</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>34</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="msscyc_2.html#E4:5_1_2_1_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="msscyc_2.html#E11:5_1_2_1_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="msscyc_2.html#E12:5_1_2_1_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="msscyc_2.html#E16:5_1_2_1_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="msscyc_1.html#T16">MSSCYC_1:16</a></i>;<br/><span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">c<sub>33</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 1 = 
the <a href="graph_1.html#U3">Source</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>29</sub></font>,<font color="Maroon">c<sub>30</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="msscyc_2.html#E13:5_1_2_1_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>29</sub></font>,<font color="Maroon">c<sub>30</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="txt" href="msscyc_2.html#E9:5_1_2_1_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="msscyc_2.html#D2" target="_self">Def2</a></i>
<br/>.= 
 <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>28</sub></font>
<b>by </b><i><a class="txt" href="msscyc_2.html#E9:5_1_2_1_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/><b>hence </b><a NAME="E22:5_1_2_1_2"/>
<span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">c<sub>34</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>28</sub></font>
 <b>by </b><i><a class="txt" href="msscyc_2.html#E11:5_1_2_1_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="msscyc_2.html#E12:5_1_2_1_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="msscyc_2.html#E16:5_1_2_1_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="msscyc_1.html#T16">MSSCYC_1:16</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E30:5_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="msscyc_2.html#E28:5_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="msscyc_2.html#E29:5_1"><i><font color="Green">E23</font></i></a>);<br/></i>
<b>then consider </b><font color="Maroon">c<sub>24</sub></font> being  <a href="graph_1.html#V8">directed</a>  <a href="graph_1.html#M1">Chain</a> of  <a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>25</sub></font> being   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E32:5_1"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="trees_2.html#K5">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>25</sub></font> )
 <b>by </b><i><a class="txt" href="msscyc_2.html#E1:5"><i><font color="Green">E6</font></i></a></i>;<br/>
<b>thus </b><a NAME="E33:5_1"/>
ex <font color="Olive">b<sub>1</sub></font> being <a href="graph_1.html#V8">directed</a>  <a href="graph_1.html#M1">Chain</a> of  <a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">c<sub>1</sub></font> st <br/>(  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="msscyc_2.html#E32:5_1"><i><font color="Green">E24</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>given </b><font color="Maroon">c<sub>6</sub></font> being  <a href="graph_1.html#V8">directed</a>  <a href="graph_1.html#M1">Chain</a> of  <a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><a NAME="E6:5"/><i><font color="Green">E9</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>7</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = the <a href="graph_1.html#U2">Edges</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="msaterm.html#K1">-Terms</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = the <a href="graph_1.html#U3">Source</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>11</sub></font> = the <a href="graph_1.html#U4">Target</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E7:5"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U2">Edges</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="msscyc_1.html#D1">MSSCYC_1:def 1</a></i>;<br/>
<a NAME="E8:5"/><i><font color="Green">E11</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="msscyc_2.html#E1:5"><i><font color="Green">E6</font></i></a>, <a class="txt" href="msscyc_2.html#E6:5"><i><font color="Green">E9</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E9:5"/><b>then </b><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="msscyc_2.html#K1" target="_self">InducedEdges</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="msscyc_2.html#E7:5"><i><font color="Green">E10</font></i></a>, <a class="ref" href="dtconstr.html#T2">DTCONSTR:2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = the <a href="graph_1.html#U2">Edges</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="msscyc_2.html#E7:5"><i><font color="Green">E10</font></i></a>, <a class="txt" href="msscyc_2.html#E8:5"><i><font color="Green">E11</font></i></a>, <a class="ref" href="dtconstr.html#T2">DTCONSTR:2</a></i>;<br/>
<a NAME="E11:5"/><i><font color="Green">E13</font></i>: 
not  <a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">c<sub>1</sub></font> is <a href="msscyc_1.html#V2">empty</a>
 
<b>by </b><i><a class="txt" href="msscyc_2.html#E9:5"><i><font color="Green">E12</font></i></a>, <a class="ref" href="msscyc_1.html#D3">MSSCYC_1:def 3</a></i>;<br/>
<a NAME="E12:5"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="msscyc_2.html#E1:5"><i><font color="Green">E6</font></i></a>, <a class="txt" href="msscyc_2.html#E6:5"><i><font color="Green">E9</font></i></a></i>;<br/>
<a NAME="E13:5"/><b>then </b><i><font color="Green">E14</font></i>: 
not <font color="Maroon">c<sub>6</sub></font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E14:5"/><i><font color="Green">E15</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="pboole.html#D16">PBOOLE:def 16</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E16:5"/><i><font color="Green">E16</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="pboole.html#S1">PBOOLE:sch 1</a>(<a class="txt" href="msscyc_2.html#E14:5"><i><font color="Green">E15</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>2</sub></font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>ex <font color="Olive">b<sub>5</sub></font> being   <a href="msaterm.html#M1">ArgumentSeq</a> of  <a href="msaterm.html#K2">Sym</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Olive">b<sub>5</sub></font> &amp; ( for <font color="Olive">b<sub>6</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">b<sub>5</sub></font> holds <br/>ex <font color="Olive">b<sub>7</sub></font> being  <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>6</sub></font> &amp;  <a href="msaterm.html#K7">the_sort_of</a> <font color="Olive">b<sub>7</sub></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="Olive">b<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>6</sub></font> &amp; (  <a href="msaterm.html#K7">the_sort_of</a> <font color="Olive">b<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp;  <a href="msaterm.html#K7">the_sort_of</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> implies <font color="Olive">b<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> ) &amp; ( (  <a href="msaterm.html#K7">the_sort_of</a> <font color="Olive">b<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> or  <a href="msaterm.html#K7">the_sort_of</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> ) implies <font color="Olive">b<sub>7</sub></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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="msaterm.html#K7">the_sort_of</a> <font color="Olive">b<sub>7</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="msaterm.html#K7">the_sort_of</a> <font color="Olive">b<sub>7</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> ) ) ) );<br/>
<a NAME="E17:5"/><i><font color="Green">E17</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> holds <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="trees_9.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="msaterm.html#K1">-Terms</a> <font color="Maroon">c<sub>2</sub></font>ex <font color="Olive">b<sub>3</sub></font> being   <a href="trees_9.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="msaterm.html#K1">-Terms</a> <font color="Maroon">c<sub>2</sub></font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:5_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<a NAME="E2:5_2"/><b>then </b><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>15</sub></font> be    <a href="trees_9.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="msaterm.html#K1">-Terms</a> <font color="Maroon">c<sub>2</sub></font>;<br/>

<a NAME="E3:5_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E4:5_2"/><b>then </b>
<font color="Maroon">c<sub>14</sub></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">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="msscyc_2.html#E6:5"><i><font color="Green">E9</font></i></a>, <a class="txt" href="msscyc_2.html#E2:5_2"><i><font color="Green">E18</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>12</sub></font> <b>by </b><i><a class="txt" href="msscyc_2.html#E7:5"><i><font color="Green">E10</font></i></a>, <a class="ref" href="dtconstr.html#T2">DTCONSTR:2</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>18</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:5_2"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> )
 <b>and </b><br/><a NAME="E8:5_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a>ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K13">*</a>  st <br/>( the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> )
 <b>by </b><i><a class="ref" href="msscyc_2.html#D1" target="_self">Def1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>17</sub></font> as   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="msscyc_2.html#E7:5_2"><i><font color="Green">E19</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>18</sub></font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="msscyc_2.html#E7:5_2"><i><font color="Green">E19</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>21</sub></font> =  <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">c<sub>19</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>22</sub></font> =  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ( ( <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> &amp;  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> ) &amp; ( ( <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>20</sub></font> or  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>20</sub></font> ) implies <font color="Maroon">a<sub>2</sub></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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> ) );<br/>
<a NAME="E11:5_2"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <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">c<sub>19</sub></font></span>)</span> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="msaterm.html#K1">-Terms</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>23</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:5_2_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <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">c<sub>19</sub></font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:5_2_1_1"/>
( ( <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> &amp;  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> ) or <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>20</sub></font> or  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>20</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_2_1_1_1"><b>suppose </b></a><a NAME="E1:5_2_1_1_1"/><i><font color="Green">E22</font></i>: 
( <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> &amp;  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> )
 ;<br/><div class="add"><b>take </b>
<font color="Maroon">c<sub>15</sub></font>
;<br/><b>thus </b><a NAME="E2:5_2_1_1_1"/>
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="msaterm.html#K1">-Terms</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>15</sub></font>] )
 <b>by </b><i><a class="txt" href="msscyc_2.html#E1:5_2_1_1_1"><i><font color="Green">E22</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_2_1_1_2"><b>suppose </b></a><a NAME="E1:5_2_1_1_2"/><i><font color="Green">E22</font></i>: 
( <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>20</sub></font> or  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>20</sub></font> )
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">c<sub>24</sub></font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="msscyc_2.html#E1:5_2_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="dtconstr.html#T2">DTCONSTR:2</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>25</sub></font> = <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <b>by </b><i><a class="txt" href="msscyc_2.html#E16:5"><i><font color="Green">E16</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>26</sub></font> =  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>24</sub></font></span><a href="tarski.html#K4">]</a></span> as   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="msaterm.html#T4">MSATERM:4</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>26</sub></font>
;<br/><b>thus </b><a NAME="E5:5_2_1_1_2"/>
( <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="msaterm.html#K1">-Terms</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>26</sub></font>] )
 <b>by </b><i><a class="txt" href="msscyc_2.html#E1:5_2_1_1_2"><i><font color="Green">E22</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>23</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <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">c<sub>19</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font> <a href="msaterm.html#K1">-Terms</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E13:5_2"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <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">c<sub>19</sub></font></span>)</span> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>(<a class="txt" href="msscyc_2.html#E11:5_2"><i><font color="Green">E20</font></i></a>);<br/></i>
<a NAME="E14:5_2"/>
 <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">c<sub>19</sub></font></span>)</span> <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> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>23</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>1</sub></font> <a href="msaterm.html#K1">-Terms</a> <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="finseq_2.html#T28">FINSEQ_2:28</a></i>;<br/>
<a NAME="E16:5_2"/><i><font color="Green">E22</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a>  <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">c<sub>19</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>25</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:5_2_2"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>24</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>26</sub></font> = <font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font> as   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="dtconstr.html#T2">DTCONSTR:2</a></i>;<br/><b>take </b><font color="Maroon">c<sub>27</sub></font> = <font color="Maroon">c<sub>26</sub></font>;<br/><b>thus </b><a NAME="E3:5_2_2"/>
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:5_2_2_1"/>
( ( <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> &amp;  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> ) or <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>20</sub></font> or  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>20</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_2_2_1_1"><b>suppose </b></a><a NAME="E1:5_2_2_1_1"/>
( <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> &amp;  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:5_2_2_1_1"/>
 <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>27</sub></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">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font>
 <b>by </b><i><a class="txt" href="msscyc_2.html#E13:5_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="msscyc_2.html#E16:5_2"><i><font color="Green">E22</font></i></a>, <a class="txt" href="msscyc_2.html#E1:5_2_2"><i><font color="Green">E23</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_2_2_1_2"><b>suppose </b></a><a NAME="E1:5_2_2_1_2"/>
( <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>20</sub></font> or  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>20</sub></font> )
 ;<br/><div class="add"><a NAME="E2:5_2_2_1_2"/><b>then </b><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>27</sub></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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="msscyc_2.html#E13:5_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="msscyc_2.html#E16:5_2"><i><font color="Green">E22</font></i></a>, <a class="txt" href="msscyc_2.html#E1:5_2_2"><i><font color="Green">E23</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>28</sub></font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="msscyc_2.html#E16:5_2"><i><font color="Green">E22</font></i></a>, <a class="txt" href="msscyc_2.html#E1:5_2_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="dtconstr.html#T2">DTCONSTR:2</a></i>;<br/><a NAME="E4:5_2_2_1_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font>
 <b>by </b><i><a class="txt" href="msscyc_2.html#E16:5"><i><font color="Green">E16</font></i></a></i>;<br/><b>hence </b><a NAME="E5:5_2_2_1_2"/>
 <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>27</sub></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">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font>
 <b>by </b><i><a class="txt" href="msscyc_2.html#E2:5_2_2_1_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="msaterm.html#T14">MSATERM:14</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>25</sub></font> = <font color="Maroon">c<sub>24</sub></font> as    <a href="msaterm.html#M1">ArgumentSeq</a> of  <a href="msaterm.html#K2">Sym</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="msscyc_2.html#E16:5_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="msaterm.html#T24">MSATERM:24</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>26</sub></font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">c<sub>25</sub></font> as   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="msaterm.html#T1">MSATERM:1</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>26</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>19</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>20</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>15</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>26</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>25</sub></font>
;<br/>

<b>thus </b><a NAME="E20:5_2"/>
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> )
 ;<br/>

<b>thus </b><a NAME="E21:5_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="msscyc_2.html#E7:5_2"><i><font color="Green">E19</font></i></a></i>;<br/>

<b>thus </b><a NAME="E22:5_2"/>
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">c<sub>25</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>27</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E23:5_2"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>25</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>28</sub></font> = <font color="Maroon">c<sub>25</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font> as   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="dtconstr.html#T2">DTCONSTR:2</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>28</sub></font>
;<br/>

<b>thus </b><a NAME="E25:5_2"/>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font>
 ;<br/>

<b>thus </b><a NAME="E26:5_2"/>
 <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>28</sub></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">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font>
 <b>by </b><i><a class="txt" href="msscyc_2.html#E23:5_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="msaterm.html#T23">MSATERM:23</a></i>;<br/>

<b>hence </b><a NAME="E27:5_2"/>
( (  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> &amp;  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> implies <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> ) &amp; ( (  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>20</sub></font> or  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>20</sub></font> ) implies <font color="Maroon">c<sub>28</sub></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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>28</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>28</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> ) )
 <b>by </b><i><a class="txt" href="msscyc_2.html#E13:5_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="msscyc_2.html#E16:5_2"><i><font color="Green">E22</font></i></a>, <a class="txt" href="msscyc_2.html#E23:5_2"><i><font color="Green">E23</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E19:5"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a>ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K13">*</a>  st <br/>( the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> ) )
 <b>by </b><i><a class="txt" href="msscyc_2.html#E9:5"><i><font color="Green">E12</font></i></a>, <a class="ref" href="msscyc_2.html#D1" target="_self">Def1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="msscyc_2.html#E19:5"><i><font color="Green">E18</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>15</sub></font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="msscyc_2.html#E19:5"><i><font color="Green">E18</font></i></a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>;<br/>
<b>consider </b><font color="Maroon">c<sub>18</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E23:5"/><i><font color="Green">E19</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>18</sub></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">c<sub>16</sub></font></span>)</span> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <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">c<sub>16</sub></font></span>)</span></span>)</span> holds <br/><font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S2">FINSEQ_1:sch 2</a>();<br/></i>
<a NAME="E24:5"/><i><font color="Green">E20</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>18</sub></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">c<sub>18</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E25:5"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>18</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="msaterm.html#K7">the_sort_of</a> <font color="Olive">b<sub>2</sub></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">c<sub>16</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>19</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:5_3"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>18</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>20</sub></font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>;<br/>
<a NAME="E2:5_3"/>
 <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">c<sub>16</sub></font></span>)</span> <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> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>22</sub></font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="msscyc_2.html#E23:5"><i><font color="Green">E19</font></i></a>, <a class="txt" href="msscyc_2.html#E24:5"><i><font color="Green">E20</font></i></a>, <a class="txt" href="msscyc_2.html#E1:5_3"><i><font color="Green">E21</font></i></a>, <a class="ref" href="dtconstr.html#T2">DTCONSTR:2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> <b>by </b><i><a class="txt" href="msscyc_2.html#E16:5"><i><font color="Green">E16</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>24</sub></font> =  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>22</sub></font></span><a href="tarski.html#K4">]</a></span> as   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="msaterm.html#T4">MSATERM:4</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>24</sub></font>
;<br/>

<b>thus </b><a NAME="E6:5_3"/>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="msscyc_2.html#E23:5"><i><font color="Green">E19</font></i></a>, <a class="txt" href="msscyc_2.html#E24:5"><i><font color="Green">E20</font></i></a>, <a class="txt" href="msscyc_2.html#E1:5_3"><i><font color="Green">E21</font></i></a></i>;<br/>

<b>thus </b><a NAME="E7:5_3"/>
 <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>24</sub></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">c<sub>16</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="ref" href="msaterm.html#T14">MSATERM:14</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</sub></font> as    <a href="msaterm.html#M1">ArgumentSeq</a> of  <a href="msaterm.html#K2">Sym</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="msscyc_2.html#E23:5"><i><font color="Green">E19</font></i></a>, <a class="ref" href="msaterm.html#T24">MSATERM:24</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>20</sub></font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">c<sub>19</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>21</sub></font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">c<sub>19</sub></font> as   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="msaterm.html#T1">MSATERM:1</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>22</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>1</sub></font> <a href="msaterm.html#K1">-Terms</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E29:5"/><i><font color="Green">E21</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; ( <font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>] ) )
 <b>from </b><i><a class="ref" href="recdef_1.html#S6">RECDEF_1:sch 6</a>(<a class="txt" href="msscyc_2.html#E17:5"><i><font color="Green">E17</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> implies ex <font color="Olive">b<sub>1</sub></font> being  <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>ex <font color="Olive">b<sub>2</sub></font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a>  &amp;  <a href="msaterm.html#K7">the_sort_of</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Olive">b<sub>2</sub></font> &amp;  <a href="trees_1.html#K6">height</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> ) );<br/>
<a NAME="E30:5"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
;<br/>
<a NAME="E31:5"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>23</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:5_4"/><i><font color="Green">E24</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> implies ex <font color="Olive">b<sub>1</sub></font> being  <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>ex <font color="Olive">b<sub>2</sub></font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a>  &amp;  <a href="msaterm.html#K7">the_sort_of</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Olive">b<sub>2</sub></font> &amp;  <a href="trees_1.html#K6">height</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> ) )
 ;<br/>

<b>assume </b><a NAME="E2:5_4"/><i><font color="Green">E25</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<a NAME="E3:5_4"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E4:5_4"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="msscyc_2.html#E2:5_4"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E5:5_4"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="msscyc_2.html#E2:5_4"><i><font color="Green">E25</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:5_4_1"/>
( <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_4_1_1"><b>suppose </b></a><a NAME="E1:5_4_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>take </b>
<font color="Maroon">c<sub>21</sub></font>
;<br/><b>take </b>
<font color="Maroon">c<sub>16</sub></font>
;<br/><b>thus </b><a NAME="E2:5_4_1_1"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="msscyc_2.html#E1:5"><i><font color="Green">E6</font></i></a>, <a class="txt" href="msscyc_2.html#E29:5"><i><font color="Green">E21</font></i></a>, <a class="txt" href="msscyc_2.html#E1:5_4_1_1"><i><font color="Green">E29</font></i></a></i>;<br/><b>thus </b><a NAME="E3:5_4_1_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a> 
 <b>by </b><i><a class="txt" href="msscyc_2.html#E19:5"><i><font color="Green">E18</font></i></a>, <a class="txt" href="msscyc_2.html#E1:5_4_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/><a NAME="E4:5_4_1_1"/>
<font color="Maroon">c<sub>21</sub></font> <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">c<sub>16</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="trees_4.html#D4">TREES_4:def 4</a></i>;<br/><b>hence </b><a NAME="E5:5_4_1_1"/>
 <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="ref" href="msaterm.html#T17">MSATERM:17</a></i>;<br/><a NAME="E6:5_4_1_1"/><i><font color="Green">E30</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a>  <a href="trees_3.html#K13">tree</a> <span class="p1">(<span class="default"><a href="trees_4.html#K11">doms</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="trees_4.html#T10">TREES_4:10</a></i>;<br/><a NAME="E7:5_4_1_1"/><i><font color="Green">E31</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="trees_4.html#K11">doms</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="ref" href="trees_3.html#T39">TREES_3:39</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>24</sub></font> being   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">c<sub>25</sub></font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K13">*</a> <b> such that </b><br/><a NAME="E9:5_4_1_1"/><i><font color="Green">E32</font></i>: 
( the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font> &amp; <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>25</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="msscyc_2.html#E19:5"><i><font color="Green">E18</font></i></a></i>;<br/><a NAME="E10:5_4_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="msscyc_2.html#E9:5_4_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="msualg_1.html#D6">MSUALG_1:def 6</a></i>;<br/><a NAME="E11:5_4_1_1"/><i><font color="Green">E34</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>25</sub></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">c<sub>25</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>26</sub></font> = <font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> as   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="msscyc_2.html#E23:5"><i><font color="Green">E19</font></i></a>, <a class="txt" href="msscyc_2.html#E24:5"><i><font color="Green">E20</font></i></a>, <a class="txt" href="msscyc_2.html#E9:5_4_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="msscyc_2.html#E10:5_4_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="msaterm.html#T22">MSATERM:22</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>27</sub></font> =  <a href="trees_4.html#K11">doms</a> <font color="Maroon">c<sub>19</sub></font> as  <a href="trees_3.html#V5">FinTree-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a> ;<br/><b>reconsider </b><font color="Maroon">c<sub>28</sub></font> =  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>26</sub></font> as  <a href="finset_1.html#V1">finite</a> <a href="trees_1.html#NM1">Tree</a> ;<br/><a NAME="E15:5_4_1_1"/>
<span class="p1">(<span class="default"><a href="trees_4.html#K11">doms</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>26</sub></font>
 <b>by </b><i><a class="txt" href="msscyc_2.html#E23:5"><i><font color="Green">E19</font></i></a>, <a class="txt" href="msscyc_2.html#E24:5"><i><font color="Green">E20</font></i></a>, <a class="txt" href="msscyc_2.html#E9:5_4_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="msscyc_2.html#E10:5_4_1_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="msscyc_2.html#E11:5_4_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_6.html#T31">FUNCT_6:31</a></i>;<br/><a NAME="E16:5_4_1_1"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>27</sub></font>
 <b>by </b><i><a class="txt" href="msscyc_2.html#E23:5"><i><font color="Green">E19</font></i></a>, <a class="txt" href="msscyc_2.html#E24:5"><i><font color="Green">E20</font></i></a>, <a class="txt" href="msscyc_2.html#E7:5_4_1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="msscyc_2.html#E9:5_4_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="msscyc_2.html#E10:5_4_1_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="msscyc_2.html#E11:5_4_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E17:5_4_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="msscyc_2.html#E23:5"><i><font color="Green">E19</font></i></a>, <a class="txt" href="msscyc_2.html#E9:5_4_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="msscyc_2.html#E10:5_4_1_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="msscyc_2.html#E11:5_4_1_1"><i><font color="Green">E34</font></i></a></i>;<br/><a NAME="E18:5_4_1_1"/><b>then </b><i><font color="Green">E37</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a>  <a href="trees_1.html#K2">elementary_tree</a> 0
 <b>by </b><i><a class="ref" href="trees_4.html#T3">TREES_4:3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_4_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>29</sub></font> be  <a href="finset_1.html#V1">finite</a> <a href="trees_1.html#NM1">Tree</a>;<br/><b>assume </b><a NAME="E1:5_4_1_1_1"/>
<font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>27</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>30</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:5_4_1_1_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>27</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>29</sub></font> )
 <b>by </b><i><a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/><a NAME="E4:5_4_1_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="msscyc_2.html#E23:5"><i><font color="Green">E19</font></i></a>, <a class="txt" href="msscyc_2.html#E24:5"><i><font color="Green">E20</font></i></a>, <a class="txt" href="msscyc_2.html#E7:5_4_1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="msscyc_2.html#E3:5_4_1_1_1"><i><font color="Green">E38</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>31</sub></font> = <font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font> as   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="msscyc_2.html#E7:5_4_1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="msscyc_2.html#E3:5_4_1_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="msaterm.html#T22">MSATERM:22</a></i>;<br/><a NAME="E6:5_4_1_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>31</sub></font>
 <b>by </b><i><a class="txt" href="msscyc_2.html#E7:5_4_1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="msscyc_2.html#E3:5_4_1_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="funct_6.html#T31">FUNCT_6:31</a></i>;<br/><a NAME="E7:5_4_1_1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a>  <a href="trees_1.html#K2">elementary_tree</a> 0
 <b>by </b><i><a class="txt" href="msscyc_2.html#E4:5_4_1_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="trees_4.html#T3">TREES_4:3</a></i>;<br/><b>hence </b><a NAME="E8:5_4_1_1_1"/>
 <a href="trees_1.html#K6">height</a> <font color="Maroon">c<sub>29</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="trees_1.html#K6">height</a> <font color="Maroon">c<sub>28</sub></font>
 <b>by </b><i><a class="txt" href="msscyc_2.html#E17:5_4_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="msscyc_2.html#E3:5_4_1_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="msscyc_2.html#E6:5_4_1_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="trees_4.html#T3">TREES_4:3</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E20:5_4_1_1"/>
 <a href="trees_1.html#K6">height</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="msscyc_2.html#E1:5_4_1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="msscyc_2.html#E6:5_4_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="msscyc_2.html#E16:5_4_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="msscyc_2.html#E18:5_4_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="trees_1.html#T79">TREES_1:79</a>, <a class="ref" href="trees_3.html#T82">TREES_3:82</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_4_1_2"><b>suppose </b></a><a NAME="E1:5_4_1_2"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:5_4_1_2"/><b>then </b><i><font color="Green">E30</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>24</sub></font> being   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>25</sub></font> being   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:5_4_1_2"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a>  &amp;  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>25</sub></font> &amp;  <a href="trees_1.html#K6">height</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> )
 <b>by </b><i><a class="txt" href="msscyc_2.html#E1:5_4"><i><font color="Green">E24</font></i></a>, <a class="txt" href="msscyc_2.html#E2:5_4"><i><font color="Green">E25</font></i></a>, <a class="txt" href="msscyc_2.html#E3:5_4"><i><font color="Green">E26</font></i></a>, <a class="txt" href="msscyc_2.html#E1:5_4_1_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>26</sub></font> being   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>27</sub></font> being   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>28</sub></font>, <font color="Maroon">c<sub>29</sub></font> being   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>30</sub></font> being    <a href="msaterm.html#M1">ArgumentSeq</a> of  <a href="msaterm.html#K2">Sym</a> <font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E6:5_4_1_2"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>29</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="tarski.html#K4">]</a></span> )
 <b>and </b><br/><a NAME="E7:5_4_1_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>26</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">c<sub>30</sub></font>
 <b>and </b><br/><a NAME="E8:5_4_1_2"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>30</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>30</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="msaterm.html#K7">the_sort_of</a> <font color="Olive">b<sub>2</sub></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">c<sub>26</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; (  <a href="msaterm.html#K7">the_sort_of</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> &amp;  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> implies <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> ) &amp; ( (  <a href="msaterm.html#K7">the_sort_of</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>27</sub></font> or  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>27</sub></font> ) implies <font color="Olive">b<sub>2</sub></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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="msaterm.html#K7">the_sort_of</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="msaterm.html#K7">the_sort_of</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> ) )
 <b>by </b><i><a class="txt" href="msscyc_2.html#E29:5"><i><font color="Green">E21</font></i></a>, <a class="txt" href="msscyc_2.html#E5:5_4"><i><font color="Green">E28</font></i></a>, <a class="txt" href="msscyc_2.html#E2:5_4_1_2"><i><font color="Green">E30</font></i></a></i>;<br/><a NAME="E9:5_4_1_2"/><i><font color="Green">E35</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>30</sub></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">c<sub>26</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="msaterm.html#T22">MSATERM:22</a></i>;<br/><a NAME="E10:5_4_1_2"/><i><font color="Green">E36</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a>  <a href="trees_3.html#K13">tree</a> <span class="p1">(<span class="default"><a href="trees_4.html#K11">doms</a> <font color="Maroon">c<sub>30</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="msscyc_2.html#E7:5_4_1_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="trees_4.html#T10">TREES_4:10</a></i>;<br/><a NAME="E11:5_4_1_2"/><i><font color="Green">E37</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="trees_4.html#K11">doms</a> <font color="Maroon">c<sub>30</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>30</sub></font>
 <b>by </b><i><a class="ref" href="trees_3.html#T39">TREES_3:39</a></i>;<br/><a NAME="E12:5_4_1_2"/><i><font color="Green">E38</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>30</sub></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">c<sub>30</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>set </b><font color="Maroon">c<sub>31</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><a NAME="E13:5_4_1_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>23</sub></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">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="msscyc_2.html#E6:5"><i><font color="Green">E9</font></i></a>, <a class="txt" href="msscyc_2.html#E2:5_4"><i><font color="Green">E25</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E14:5_4_1_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="msscyc_2.html#E7:5"><i><font color="Green">E10</font></i></a>, <a class="ref" href="dtconstr.html#T2">DTCONSTR:2</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>32</sub></font>, <font color="Maroon">c<sub>33</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E16:5_4_1_2"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>33</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a>ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K13">*</a>  st <br/>( the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>33</sub></font> ) )
 <b>by </b><i><a class="ref" href="msscyc_2.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E17:5_4_1_2"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="msscyc_2.html#E6:5"><i><font color="Green">E9</font></i></a>, <a class="txt" href="msscyc_2.html#E4:5_4"><i><font color="Green">E27</font></i></a>, <a class="txt" href="msscyc_2.html#E2:5_4_1_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>34</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>12</sub></font> <b>by </b><i><a class="txt" href="msscyc_2.html#E7:5"><i><font color="Green">E10</font></i></a>, <a class="ref" href="dtconstr.html#T2">DTCONSTR:2</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>35</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>12</sub></font> <b>by </b><i><a class="txt" href="msscyc_2.html#E7:5"><i><font color="Green">E10</font></i></a>, <a class="txt" href="msscyc_2.html#E13:5_4_1_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="dtconstr.html#T2">DTCONSTR:2</a></i>;<br/><i><font color="Green">E41</font></i>:  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>24</sub></font> = 
the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>34</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span>
<b>by </b><i><a class="txt" href="msscyc_2.html#E4:5_4_1_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="msualg_1.html#D7">MSUALG_1:def 7</a></i>
<br/>.= 
the <a href="graph_1.html#U4">Target</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>34</sub></font>
<b>by </b><i><a class="ref" href="msscyc_2.html#D3" target="_self">Def3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="msscyc_2.html#E6:5"><i><font color="Green">E9</font></i></a>, <a class="txt" href="msscyc_2.html#E11:5"><i><font color="Green">E13</font></i></a>, <a class="txt" href="msscyc_2.html#E13:5"><i><font color="Green">E14</font></i></a>, <a class="txt" href="msscyc_2.html#E4:5_4"><i><font color="Green">E27</font></i></a>, <a class="txt" href="msscyc_2.html#E2:5_4_1_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="msscyc_1.html#T11">MSSCYC_1:11</a></i>
<br/>.= 
the <a href="graph_1.html#U3">Source</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>35</sub></font>
<b>by </b><i><a class="txt" href="msscyc_2.html#E6:5"><i><font color="Green">E9</font></i></a>, <a class="txt" href="msscyc_2.html#E11:5"><i><font color="Green">E13</font></i></a>, <a class="txt" href="msscyc_2.html#E13:5"><i><font color="Green">E14</font></i></a>, <a class="txt" href="msscyc_2.html#E2:5_4"><i><font color="Green">E25</font></i></a>, <a class="ref" href="msscyc_1.html#T11">MSSCYC_1:11</a></i>
<br/>.= 
<font color="Maroon">c<sub>35</sub></font> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="ref" href="msscyc_2.html#D2" target="_self">Def2</a></i>
<br/>.= 
<font color="Maroon">c<sub>27</sub></font>
<b>by </b><i><a class="txt" href="msscyc_2.html#E6:5_4_1_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/><a NAME="E21:5_4_1_2"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>32</sub></font> &amp; <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>33</sub></font> )
 <b>by </b><i><a class="txt" href="msscyc_2.html#E6:5_4_1_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="msscyc_2.html#E16:5_4_1_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>36</sub></font> being   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">c<sub>37</sub></font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K13">*</a> <b> such that </b><br/><a NAME="E23:5_4_1_2"/><i><font color="Green">E43</font></i>: 
( the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>37</sub></font> &amp; <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>37</sub></font> &amp; <font color="Maroon">c<sub>37</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> )
 <b>by </b><i><a class="txt" href="msscyc_2.html#E16:5_4_1_2"><i><font color="Green">E40</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>38</sub></font> = <font color="Maroon">c<sub>32</sub></font> as   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="msscyc_2.html#E16:5_4_1_2"><i><font color="Green">E40</font></i></a></i>;<br/><a NAME="E25:5_4_1_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>37</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>38</sub></font>
 <b>by </b><i><a class="txt" href="msscyc_2.html#E23:5_4_1_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="msualg_1.html#D6">MSUALG_1:def 6</a></i>;<br/><a NAME="E26:5_4_1_2"/><i><font color="Green">E45</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>37</sub></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">c<sub>37</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>39</sub></font> being   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E28:5_4_1_2"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>39</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>30</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font> &amp;  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>39</sub></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">c<sub>26</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font> &amp; (  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>39</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> &amp;  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> implies <font color="Maroon">c<sub>39</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> ) &amp; ( (  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>39</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>27</sub></font> or  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>27</sub></font> ) implies <font color="Maroon">c<sub>39</sub></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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>39</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>39</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> ) )
 <b>by </b><i><a class="txt" href="msscyc_2.html#E8:5_4_1_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="msscyc_2.html#E9:5_4_1_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="msscyc_2.html#E12:5_4_1_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="msscyc_2.html#E21:5_4_1_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="msscyc_2.html#E23:5_4_1_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="msscyc_2.html#E25:5_4_1_2"><i><font color="Green">E44</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>40</sub></font> =  <a href="trees_4.html#K11">doms</a> <font color="Maroon">c<sub>30</sub></font> as  <a href="trees_3.html#V5">FinTree-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a> ;<br/><b>reconsider </b><font color="Maroon">c<sub>41</sub></font> =  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>39</sub></font> as  <a href="finset_1.html#V1">finite</a> <a href="trees_1.html#NM1">Tree</a> ;<br/><a NAME="E31:5_4_1_2"/>
<span class="p1">(<span class="default"><a href="trees_4.html#K11">doms</a> <font color="Maroon">c<sub>30</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>39</sub></font>
 <b>by </b><i><a class="txt" href="msscyc_2.html#E9:5_4_1_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="msscyc_2.html#E12:5_4_1_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="msscyc_2.html#E21:5_4_1_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="msscyc_2.html#E23:5_4_1_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="msscyc_2.html#E25:5_4_1_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="msscyc_2.html#E26:5_4_1_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="msscyc_2.html#E28:5_4_1_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="funct_6.html#T31">FUNCT_6:31</a></i>;<br/><a NAME="E32:5_4_1_2"/><b>then </b><i><font color="Green">E47</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>39</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>40</sub></font>
 <b>by </b><i><a class="txt" href="msscyc_2.html#E9:5_4_1_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="msscyc_2.html#E11:5_4_1_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="msscyc_2.html#E12:5_4_1_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="msscyc_2.html#E21:5_4_1_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="msscyc_2.html#E23:5_4_1_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="msscyc_2.html#E25:5_4_1_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="msscyc_2.html#E26:5_4_1_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><div><i><font color="Green">E48</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="5_4_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>42</sub></font> be  <a href="finset_1.html#V1">finite</a> <a href="trees_1.html#NM1">Tree</a>;<br/><b>assume </b><a NAME="E1:5_4_1_2_2"/>
<font color="Maroon">c<sub>42</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>40</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>43</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:5_4_1_2_2"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>40</sub></font> &amp; <font color="Maroon">c<sub>40</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>42</sub></font> )
 <b>by </b><i><a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>44</sub></font> being   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E5:5_4_1_2_2"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">c<sub>44</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>30</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>43</sub></font> &amp;  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>44</sub></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">c<sub>26</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>43</sub></font> &amp; (  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>44</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> &amp;  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> implies <font color="Maroon">c<sub>44</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> ) &amp; ( (  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>44</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>27</sub></font> or  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>27</sub></font> ) implies <font color="Maroon">c<sub>44</sub></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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>44</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>44</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> ) )
 <b>by </b><i><a class="txt" href="msscyc_2.html#E8:5_4_1_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="msscyc_2.html#E11:5_4_1_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="msscyc_2.html#E3:5_4_1_2_2"><i><font color="Green">E49</font></i></a></i>;<br/><a NAME="E6:5_4_1_2_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>40</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>44</sub></font>
 <b>by </b><i><a class="txt" href="msscyc_2.html#E11:5_4_1_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="msscyc_2.html#E3:5_4_1_2_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="msscyc_2.html#E5:5_4_1_2_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="funct_6.html#T31">FUNCT_6:31</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_4_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:5_4_1_2_2_1"/>
( (  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>44</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> &amp;  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> ) or  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>44</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>27</sub></font> or  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>27</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_4_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:5_4_1_2_2_1_1"/>
(  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>44</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> &amp;  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:5_4_1_2_2_1_1"/>
 <a href="trees_1.html#K6">height</a> <font color="Maroon">c<sub>42</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="trees_1.html#K6">height</a> <font color="Maroon">c<sub>41</sub></font>
 <b>by </b><i><a class="txt" href="msscyc_2.html#E11:5_4_1_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="msscyc_2.html#E21:5_4_1_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="msscyc_2.html#E23:5_4_1_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="msscyc_2.html#E28:5_4_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="msscyc_2.html#E3:5_4_1_2_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="msscyc_2.html#E5:5_4_1_2_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="funct_6.html#T31">FUNCT_6:31</a>, <a class="ref" href="msualg_1.html#D6">MSUALG_1:def 6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_4_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:5_4_1_2_2_1_2"/>
(  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>44</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>27</sub></font> or  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>27</sub></font> )
 ;<br/><div class="add"><a NAME="E2:5_4_1_2_2_1_2"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>44</sub></font> <a href="hidden.html#R1">=</a>  <a href="trees_1.html#K2">elementary_tree</a> 0
 <b>by </b><i><a class="txt" href="msscyc_2.html#E5:5_4_1_2_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="trees_4.html#T3">TREES_4:3</a></i>;<br/><b>hence </b><a NAME="E3:5_4_1_2_2_1_2"/>
 <a href="trees_1.html#K6">height</a> <font color="Maroon">c<sub>42</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="trees_1.html#K6">height</a> <font color="Maroon">c<sub>41</sub></font>
 <b>by </b><i><a class="txt" href="msscyc_2.html#E3:5_4_1_2_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="msscyc_2.html#E6:5_4_1_2_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="trees_1.html#T79">TREES_1:79</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>take </b>
<font color="Maroon">c<sub>29</sub></font>
;<br/><b>take </b>
<font color="Maroon">c<sub>26</sub></font>
;<br/><b>thus </b><a NAME="E34:5_4_1_2"/>
<font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="msscyc_2.html#E6:5_4_1_2"><i><font color="Green">E32</font></i></a></i>;<br/><b>thus </b><a NAME="E35:5_4_1_2"/>
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a> 
 <b>by </b><i><a class="txt" href="msscyc_2.html#E6:5_4_1_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/><a NAME="E36:5_4_1_2"/>
<font color="Maroon">c<sub>29</sub></font> <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">c<sub>26</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="msscyc_2.html#E7:5_4_1_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="trees_4.html#D4">TREES_4:def 4</a></i>;<br/><b>hence </b><a NAME="E37:5_4_1_2"/>
 <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>26</sub></font>
 <b>by </b><i><a class="ref" href="msaterm.html#T17">MSATERM:17</a></i>;<br/><b>thus </b><a NAME="E38:5_4_1_2"/>
 <a href="trees_1.html#K6">height</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="msscyc_2.html#E4:5_4_1_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="msscyc_2.html#E6:5_4_1_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="msscyc_2.html#E10:5_4_1_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="msscyc_2.html#E20:5_4_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="msscyc_2.html#E21:5_4_1_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="msscyc_2.html#E23:5_4_1_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="msscyc_2.html#E28:5_4_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="msscyc_2.html#E32:5_4_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="msscyc_2.html#E33:5_4_1_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="msualg_1.html#D6">MSUALG_1:def 6</a>, <a class="ref" href="trees_3.html#T82">TREES_3:82</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E32:5"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="msscyc_2.html#E30:5"><i><font color="Green">E22</font></i></a>, <a class="txt" href="msscyc_2.html#E31:5"><i><font color="Green">E23</font></i></a>);<br/></i>
<b>then consider </b><font color="Maroon">c<sub>23</sub></font> being   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>24</sub></font> being   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E34:5"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a>  &amp;  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>24</sub></font> &amp;  <a href="trees_1.html#K6">height</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="msscyc_2.html#E1:5"><i><font color="Green">E6</font></i></a></i>;<br/>
<a NAME="E35:5"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="msscyc_2.html#E1:5"><i><font color="Green">E6</font></i></a>, <a class="txt" href="msscyc_2.html#E6:5"><i><font color="Green">E9</font></i></a></i>;<br/>
<a NAME="E36:5"/><b>then </b><i><font color="Green">E25</font></i>: 
not <font color="Maroon">c<sub>6</sub></font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E37:5"/><i><font color="Green">E26</font></i>: 
not  <a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">c<sub>1</sub></font> is <a href="msscyc_1.html#V2">empty</a>
 
<b>by </b><i><a class="txt" href="msscyc_2.html#E9:5"><i><font color="Green">E12</font></i></a>, <a class="ref" href="msscyc_1.html#D3">MSSCYC_1:def 3</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>25</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>;<br/>
<a NAME="E38:5"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="msscyc_2.html#E6:5"><i><font color="Green">E9</font></i></a>, <a class="txt" href="msscyc_2.html#E36:5"><i><font color="Green">E25</font></i></a>, <a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/>
<a NAME="E39:5"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="msscyc_2.html#K1" target="_self">InducedEdges</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="msscyc_2.html#E6:5"><i><font color="Green">E9</font></i></a>, <a class="txt" href="msscyc_2.html#E7:5"><i><font color="Green">E10</font></i></a>, <a class="ref" href="dtconstr.html#T2">DTCONSTR:2</a></i>;<br/>
<span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
the <a href="graph_1.html#U4">Target</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="msscyc_2.html#E36:5"><i><font color="Green">E25</font></i></a>, <a class="txt" href="msscyc_2.html#E37:5"><i><font color="Green">E26</font></i></a>, <a class="ref" href="msscyc_1.html#T14">MSSCYC_1:14</a></i>
<br/>.= 
the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span>
<b>by </b><i><a class="txt" href="msscyc_2.html#E39:5"><i><font color="Green">E27</font></i></a>, <a class="ref" href="msscyc_2.html#D3" target="_self">Def3</a></i>
<br/>.= 
 <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>24</sub></font>
<b>by </b><i><a class="txt" href="msscyc_2.html#E6:5"><i><font color="Green">E9</font></i></a>, <a class="txt" href="msscyc_2.html#E34:5"><i><font color="Green">E24</font></i></a>, <a class="ref" href="msualg_1.html#D7">MSUALG_1:def 7</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>26</sub></font> = <font color="Maroon">c<sub>23</sub></font> as    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="msscyc_2.html#E4:5"><i><font color="Green">E8</font></i></a>, <a class="txt" href="msscyc_2.html#E6:5"><i><font color="Green">E9</font></i></a>, <a class="txt" href="msscyc_2.html#E34:5"><i><font color="Green">E24</font></i></a>, <a class="ref" href="msaterm.html#D5">MSATERM:def 5</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>26</sub></font>
;<br/>

<b>thus </b><a NAME="E42:5"/>
 <a href="msafree2.html#K9">depth</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="msscyc_2.html#E34:5"><i><font color="Green">E24</font></i></a>, <a class="ref" href="msafree2.html#D14">MSAFREE2:def 14</a></i>;<br/>


</div>
