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<b>let </b><font color="Maroon" title="c1">S</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> non <a href="msualg_1.html#V2" title="MSUALG_1:attr.2">void</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ;<br/>

<b>set </b><font color="Maroon" title="c2">G</font> =  <a href="msscyc_2.html#K4" title="MSSCYC_2:func.4">InducedGraph</a> <font color="Maroon" title="c1">S</font>;<br/>
<b>assume </b><a NAME="E1:7"/><i><font color="Green" title="E6">A1</font></i>: 
<font color="Maroon" title="c1">S</font> is <a href="msafree2.html#V5" title="MSAFREE2:attr.5">monotonic</a>
 ;<br/>

<b>assume </b><a NAME="E2:7"/>
not  <a href="msscyc_2.html#K4" title="MSSCYC_2:func.4">InducedGraph</a> <font color="Maroon" title="c1">S</font> is <a href="msscyc_1.html#V4" title="MSSCYC_1:attr.4">well-founded</a>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c3">v</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <span class="p1">(<span class="default"><a href="msscyc_2.html#K4" title="MSSCYC_2:func.4">InducedGraph</a> <font color="Maroon" title="c1">S</font></span>)</span><b> such that </b><br/><a NAME="E4:7"/><i><font color="Green" title="E7">A2</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   ex <font color="Olive" title="b2">c</font> being <a href="msscyc_1.html#NV1" title="MSSCYC_1:NV.1">directed</a>  <a href="graph_1.html#M1" title="GRAPH_1:mode.1">Chain</a> of  <a href="msscyc_2.html#K4" title="MSSCYC_2:func.4">InducedGraph</a> <font color="Maroon" title="c1">S</font> st <br/>( not <font color="Olive" title="b2">c</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <span class="p1">(<span class="default"><a href="graph_2.html#K7" title="GRAPH_2:func.7">vertex-seq</a> <font color="Olive" title="b2">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">c</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">n</font> )
 <b>by </b><i><a class="ref" href="msscyc_1.html#D5" title="MSSCYC_1:def.5">MSSCYC_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">S</font> = <font color="Maroon" title="c1">S</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> non <a href="msualg_1.html#V2" title="MSUALG_1:attr.2">void</a> <a href="msafree2.html#V5" title="MSAFREE2:attr.5">monotonic</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a>  <b>by </b><i><a class="txt" href="msscyc_2.html#E1:7"><i><font color="Green" title="E6">A1</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">v</font> = <font color="Maroon" title="c3">v</font> as   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c4">S</font> ;<br/>
<b>consider </b><font color="Maroon" title="c6">A</font> being  <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a> <a href="msafree2.html#V4" title="MSAFREE2:attr.4">locally-finite</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c4">S</font>;<br/>
<b>consider </b><font color="Maroon" title="c7">s</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E9:7"/><i><font color="Green" title="E8">A3</font></i>: 
( <font color="Maroon" title="c7">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="circuit1.html#K5" title="CIRCUIT1:func.5">depth</a> <font color="Olive" title="b1">t</font></span>)</span> where <font color="Olive" title="b1">t</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6" title="MSAFREE2:func.6">FreeEnv</a> <font color="Maroon" title="c6">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">v</font> : verum </span>}</span>  &amp;  <a href="circuit1.html#K6" title="CIRCUIT1:func.6">depth</a> <font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c7">s</font> )
 <b>by </b><i><a class="ref" href="circuit1.html#D6" title="CIRCUIT1:def.6">CIRCUIT1:def 6</a></i>;<br/>
<a NAME="E10:7"/>
 <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c7">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">c</font> being  <a href="msscyc_1.html#NV1" title="MSSCYC_1:NV.1">directed</a>  <a href="graph_1.html#M1" title="GRAPH_1:mode.1">Chain</a> of  <a href="msscyc_2.html#K4" title="MSSCYC_2:func.4">InducedGraph</a> <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E12:7"/><i><font color="Green" title="E9">A4</font></i>: 
( not <font color="Maroon" title="c8">c</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <span class="p1">(<span class="default"><a href="graph_2.html#K7" title="GRAPH_2:func.7">vertex-seq</a> <font color="Maroon" title="c8">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">c</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c7">s</font> )
 <b>by </b><i><a class="txt" href="msscyc_2.html#E4:7"><i><font color="Green" title="E7">A2</font></i></a></i>;<br/>
<a NAME="E13:7"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">c</font>
 
<b>by </b><i><a class="txt" href="msscyc_2.html#E12:7"><i><font color="Green" title="E9">A4</font></i></a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">t</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c6">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">v</font><b> such that </b><br/><a NAME="E15:7"/><i><font color="Green" title="E10">A5</font></i>: 
 <a href="msafree2.html#K9" title="MSAFREE2:func.9">depth</a> <font color="Maroon" title="c9">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">c</font>
 <b>by </b><i><a class="txt" href="msscyc_2.html#E12:7"><i><font color="Green" title="E9">A4</font></i></a>, <a class="ref" href="msscyc_2.html#T2" target="_self" title="MSSCYC_2:th.2">Th2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">t'</font> = <font color="Maroon" title="c9">t</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6" title="MSAFREE2:func.6">FreeEnv</a> <font color="Maroon" title="c6">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">v</font> ;<br/>
<b>consider </b><font color="Maroon" title="c11">t''</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c6">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">v</font><b> such that </b><br/><a NAME="E18:7"/><i><font color="Green" title="E11">A6</font></i>: 
( <font color="Maroon" title="c10">t'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">t''</font> &amp;  <a href="circuit1.html#K5" title="CIRCUIT1:func.5">depth</a> <font color="Maroon" title="c10">t'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msafree2.html#K9" title="MSAFREE2:func.9">depth</a> <font color="Maroon" title="c11">t''</font> )
 <b>by </b><i><a class="ref" href="circuit1.html#D5" title="CIRCUIT1:def.5">CIRCUIT1:def 5</a></i>;<br/>
<a NAME="E19:7"/>
 <a href="circuit1.html#K5" title="CIRCUIT1:func.5">depth</a> <font color="Maroon" title="c10">t'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">s</font>
 
<b>by </b><i><a class="txt" href="msscyc_2.html#E9:7"><i><font color="Green" title="E8">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E20:7"/>
contradiction
 <b>by </b><i><a class="txt" href="msscyc_2.html#E12:7"><i><font color="Green" title="E9">A4</font></i></a>, <a class="txt" href="msscyc_2.html#E15:7"><i><font color="Green" title="E10">A5</font></i></a>, <a class="txt" href="msscyc_2.html#E18:7"><i><font color="Green" title="E11">A6</font></i></a>, <a class="ref" href="pre_circ.html#D1" title="PRE_CIRC:def.1">PRE_CIRC:def 1</a></i>;<br/>


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