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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> <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>set </b><font color="Maroon" title="c3">E</font> =  <a href="msscyc_2.html#K1" title="MSSCYC_2:func.1">InducedEdges</a> <font color="Maroon" title="c1">S</font>;<br/>
<b>set </b><font color="Maroon" title="c4">Sou</font> =  <a href="msscyc_2.html#K2" title="MSSCYC_2:func.2">InducedSource</a> <font color="Maroon" title="c1">S</font>;<br/>
<b>set </b><font color="Maroon" title="c5">T</font> =  <a href="msscyc_2.html#K3" title="MSSCYC_2:func.3">InducedTarget</a> <font color="Maroon" title="c1">S</font>;<br/>
<b>set </b><font color="Maroon" title="c6">OS</font> = the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c1">S</font>;<br/>
<b>set </b><font color="Maroon" title="c7">CA</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>;<br/>
<b>set </b><font color="Maroon" title="c8">AR</font> = the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c1">S</font>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:6_1"/>
<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="E2:6_1"/>
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="c9">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="E4:6_1"/><i><font color="Green" title="E6">A1</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">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>consider </b><font color="Maroon" title="c10">e</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="E6:6_1"/><i><font color="Green" title="E7">A2</font></i>: 
( not <font color="Maroon" title="c10">e</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="c10">e</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="c10">e</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">e</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 <b>by </b><i><a class="txt" href="msscyc_2.html#E4:6_1"><i><font color="Green" title="E6">A1</font></i></a></i>;<br/><a NAME="E7:6_1"/><i><font color="Green" title="E8">A3</font></i>: 
<font color="Maroon" title="c10">e</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</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>by </b><i><a class="ref" href="msscyc_1.html#D1" title="MSSCYC_1:def.1">MSSCYC_1:def 1</a></i>;<br/><a NAME="E8:6_1"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">e</font>
 <b>by </b><i><a class="txt" href="msscyc_2.html#E6:6_1"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></i>;<br/><a NAME="E9:6_1"/><b>then </b>
( <font color="Maroon" title="c10">e</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">e</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">e</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</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>by </b><i><a class="txt" href="msscyc_2.html#E7:6_1"><i><font color="Green" title="E8">A3</font></i></a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E10:6_1"/><b>then </b>
 ex <font color="Olive" title="b1">op</font>, <font color="Olive" title="b2">v</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon" title="c10">e</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">op</font>,<font color="Olive" title="b2">v</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b1">op</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c1">S</font> &amp; <font color="Olive" title="b2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> &amp;  ex <font color="Olive" title="b3">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ex <font color="Olive" title="b4">args</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a>  st <br/>( the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c1">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">op</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">args</font> &amp; <font color="Olive" title="b3">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b4">args</font> &amp; <font color="Olive" title="b4">args</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v</font> ) )
 <b>by </b><i><a class="ref" href="msscyc_2.html#D1" target="_self" title="MSSCYC_2:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E11:6_1"/>
contradiction
 <b>by </b><i><a class="ref" href="msualg_1.html#D5" title="MSUALG_1:def.5">MSUALG_1:def 5</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:6"/>
 <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>let </b><font color="Maroon" title="c9">A</font> be  <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a> <a href="msafree2.html#V3" title="MSAFREE2:attr.3">finitely-generated</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="msafree2.html#D13" title="MSAFREE2:def.13">MSAFREE2:def 13</a><br/>

<b>thus </b><a NAME="E3:6"/>
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c9">A</font> is <a href="finset_1.html#V2" title="FINSET_1:attr.2">locally-finite</a>
 <b>by </b><i><a class="ref" href="msafree2.html#D10" title="MSAFREE2:def.10">MSAFREE2:def 10</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="msafree2.html#D11" title="MSAFREE2:def.11">MSAFREE2:def 11</a><br/>


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