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

<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/>

<b>set </b><font color="Maroon">G</font> =  <a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">S</font>;<br/>
<b>assume </b><a NAME="E1:7"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">S</font> is <a href="msafree2.html#V5">monotonic</a>
 ;<br/>

<b>assume </b><a NAME="E2:7"/>
not  <a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">S</font> is <a href="msscyc_1.html#V4">well-founded</a>
 ;<br/>

<b>then consider </b><font color="Maroon">v</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U1">Vertices</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">S</font></span>)</span><b> such that </b><br/><a NAME="E4:7"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">c</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">S</font> st <br/>( not <font color="Olive">c</font> is <a href="xboole_0.html#V1">empty</a> &amp; <span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Olive">c</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">c</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">c</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Olive">n</font> )
 <b>by </b><i><a class="ref" href="msscyc_1.html#D5">MSSCYC_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">S</font> = <font color="Maroon">S</font> as  non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a> <a href="msafree2.html#V5">monotonic</a>  <a href="msualg_1.html#L1">ManySortedSign</a>  <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">v</font> = <font color="Maroon">v</font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font> ;<br/>
<b>consider </b><font color="Maroon">A</font> being  <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V4">locally-finite</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font>;<br/>
<b>consider </b><font color="Maroon">s</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E9:7"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="circuit1.html#K5">depth</a> <font color="Olive">t</font></span>)</span> where <font color="Olive">t</font> is    <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="msafree2.html#K6">FreeEnv</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> : verum </span>}</span>  &amp;  <a href="circuit1.html#K6">depth</a> <font color="Maroon">v</font>,<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="pre_circ.html#K1">max</a> <font color="Maroon">s</font> )
 <b>by </b><i><a class="ref" href="circuit1.html#D6">CIRCUIT1:def 6</a></i>;<br/>
<a NAME="E10:7"/>
 <a href="pre_circ.html#K1">max</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>then consider </b><font color="Maroon">c</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">S</font><b> such that </b><br/><a NAME="E12:7"/><i><font color="Green">E55</font></i>: 
( not <font color="Maroon">c</font> is <a href="xboole_0.html#V1">empty</a> &amp; <span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">c</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</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="pre_circ.html#K1">max</a> <font color="Maroon">s</font> )
 <b>by </b><i/>;<br/>
<a NAME="E13:7"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<b>then consider </b><font color="Maroon">t</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> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font><b> such that </b><br/><a NAME="E15:7"/><i><font color="Green">E56</font></i>: 
 <a href="msafree2.html#K9">depth</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font>
 <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">t'</font> = <font color="Maroon">t</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="msafree2.html#K6">FreeEnv</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> ;<br/>
<b>consider </b><font color="Maroon">t''</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> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font><b> such that </b><br/><a NAME="E18:7"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">t'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t''</font> &amp;  <a href="circuit1.html#K5">depth</a> <font color="Maroon">t'</font> <a href="hidden.html#R1">=</a>  <a href="msafree2.html#K9">depth</a> <font color="Maroon">t''</font> )
 <b>by </b><i><a class="ref" href="circuit1.html#D5">CIRCUIT1:def 5</a></i>;<br/>
<a NAME="E19:7"/>
 <a href="circuit1.html#K5">depth</a> <font color="Maroon">t'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">s</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E20:7"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/>


</div>
