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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="msualg_1.html#V2">void</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="msscyc_2.html#K4" target="_self">InducedGraph</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="msscyc_2.html#K1" target="_self">InducedEdges</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="msscyc_2.html#K2" target="_self">InducedSource</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="msscyc_2.html#K3" target="_self">InducedTarget</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>1</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="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:6_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="msafree2.html#V5">monotonic</a>
 ;<br/><b>assume </b><a NAME="E2:6_1"/>
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#V4">well-founded</a>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>9</sub></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">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E4:6_1"/><i><font color="Green">E6</font></i>: 
for <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="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/>( not <font color="Olive">b<sub>2</sub></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">b<sub>2</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>2</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>9</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="msscyc_1.html#D5">MSSCYC_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>10</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><br/><a NAME="E6:6_1"/><i><font color="Green">E7</font></i>: 
( not <font color="Maroon">c<sub>10</sub></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<sub>10</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>10</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>9</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 <b>by </b><i><a class="txt" href="msscyc_2.html#E4:6_1"><i><font color="Green">E6</font></i></a></i>;<br/><a NAME="E7:6_1"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>10</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="E8:6_1"/>
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#E6:6_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/><a NAME="E9:6_1"/><b>then </b>
( <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>10</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</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#E7:6_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E10:6_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">c<sub>10</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="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>1</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="Olive">b<sub>2</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>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a>ex <font color="Olive">b<sub>4</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="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">b<sub>4</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> ) )
 <b>by </b><i><a class="ref" href="msscyc_2.html#D1" target="_self">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">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" target="_self">InducedGraph</a> <font color="Maroon">c<sub>1</sub></font> is <a href="msscyc_1.html#V4">well-founded</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>9</sub></font> be  <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="msafree2.html#D13">MSAFREE2:def 13</a><br/>

<b>thus </b><a NAME="E3:6"/>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>9</sub></font> is <a href="pre_circ.html#V1">locally-finite</a>
 <b>by </b><i><a class="ref" href="msafree2.html#D10">MSAFREE2:def 10</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="msafree2.html#D11">MSAFREE2:def 11</a><br/>


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