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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="struct_0.html#V11" title="STRUCT_0:attr.11">void</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">S</font> is <a href="msscyc_1.html#V4" title="MSSCYC_1:attr.4">finitely_operated</a> &amp;  <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#V3" title="MSSCYC_1:attr.3">well-founded</a> implies <font color="Maroon" title="c1">S</font> is <a href="msafree2.html#V5" title="MSAFREE2:attr.5">monotonic</a> )</span><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="E1:9"/><i><font color="Green" title="E7">A1</font></i>: 
( <font color="Maroon" title="c1">S</font> is <a href="msscyc_1.html#V4" title="MSSCYC_1:attr.4">finitely_operated</a> &amp;  <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#V3" title="MSSCYC_1:attr.3">well-founded</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">S</font> is <a href="msafree2.html#V5" title="MSAFREE2:attr.5">monotonic</a></span><br/>

<b>given </b><font color="Maroon" title="c3">A</font> being  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">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><b> such that </b><a NAME="E3:9"/><i><font color="Green" title="E8">A2</font></i>: 
not <font color="Maroon" title="c3">A</font> is <a href="msafree2.html#V4" title="MSAFREE2:attr.4">locally-finite</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="msafree2.html#D13" title="MSAFREE2:def.13">MSAFREE2:def 13</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>consider </b><font color="Maroon" title="c4">GS</font> being  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V2</a> <a href="finset_1.html#V2" title="FINSET_1:attr.2">V26</a>  <a href="msafree.html#M1" title="MSAFREE:mode.1">GeneratorSet</a> of <font color="Maroon" title="c3">A</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">gs</font> = <font color="Maroon" title="c4">GS</font> as  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V2</a> <a href="finset_1.html#V2" title="FINSET_1:attr.2">V26</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> ;<br/>
<a NAME="E6:9"/>
not  <a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c5">gs</font> is <a href="msafree2.html#V4" title="MSAFREE2:attr.4">locally-finite</a>
 
<b>by </b><i><a class="txt" href="#E3:9"><i><font color="Green" title="E8">A2</font></i></a>, <a class="ref" href="msscyc_1.html#T23" title="MSSCYC_1:th.23">MSSCYC_1:23</a></i>;<br/>
<a NAME="E7:9"/><b>then </b>
not the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c5">gs</font></span>)</span> 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#D11" title="MSAFREE2:def.11">MSAFREE2:def 11</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">v</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E9:9"/><i><font color="Green" title="E9">A3</font></i>: 
( <font color="Maroon" title="c6">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; not the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c5">gs</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 <b>by </b><i><a class="ref" href="pre_circ.html#D3" title="PRE_CIRC:def.3">PRE_CIRC:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">v</font> = <font color="Maroon" title="c6">v</font> as   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S</font> <b>by </b><i><a class="txt" href="#E9:9"><i><font color="Green" title="E9">A3</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">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> <b> such that </b><br/><a NAME="E12:9"/><i><font color="Green" title="E10">A4</font></i>: 
for <font color="Olive" title="b1">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 not <font color="Olive" title="b1">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="b1">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="b1">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="c7">v</font> holds <br/> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">n</font>
 <b>by </b><i><a class="txt" href="#E1:9"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="msscyc_1.html#D5" title="MSSCYC_1:def.5">MSSCYC_1:def 5</a></i>;<br/>
<b>set </b><font color="Maroon" title="c9">V</font> = the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c5">gs</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">v</font>;<br/>
<b>set </b><font color="Maroon" title="c10">Vn</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">t</font> 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#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c5">gs</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">v</font> :  <a href="msafree2.html#K9" title="MSAFREE2:func.9">depth</a> <font color="Olive" title="b1">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">n</font> </span>}</span> ;<br/>
<a NAME="E13:9"/><i><font color="Green" title="E11">A5</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">t</font> 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#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c5">gs</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">v</font> :  <a href="msafree2.html#K9" title="MSAFREE2:func.9">depth</a> <font color="Olive" title="b1">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">n</font> </span>}</span>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="txt" href="#E1:9"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="msscyc_2.html#T5" target="_self" title="MSSCYC_2:th.5">Th5</a></i>;<br/>
<a NAME="E14:9"/>
not the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c5">gs</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">v</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">t</font> 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#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c5">gs</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">v</font> :  <a href="msafree2.html#K9" title="MSAFREE2:func.9">depth</a> <font color="Olive" title="b1">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">n</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="#E9:9"><i><font color="Green" title="E9">A3</font></i></a>, <a class="txt" href="#E13:9"><i><font color="Green" title="E11">A5</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">t</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E16:9"/><i><font color="Green" title="E12">A6</font></i>: 
( <font color="Maroon" title="c11">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c5">gs</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">v</font> &amp; not <font color="Maroon" title="c11">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">t</font> 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#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c5">gs</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">v</font> :  <a href="msafree2.html#K9" title="MSAFREE2:func.9">depth</a> <font color="Olive" title="b1">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">n</font> </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">t</font> = <font color="Maroon" title="c11">t</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c5">gs</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">v</font> <b>by </b><i><a class="txt" href="#E16:9"><i><font color="Green" title="E12">A6</font></i></a></i>;<br/>
<a NAME="E18:9"/><i><font color="Green" title="E13">A7</font></i>: 
not  <a href="msafree2.html#K9" title="MSAFREE2:func.9">depth</a> <font color="Maroon" title="c12">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">n</font>
 
<b>by </b><i><a class="txt" href="#E16:9"><i><font color="Green" title="E12">A6</font></i></a></i>;<br/>
<a NAME="E19:9"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="msafree2.html#K9" title="MSAFREE2:func.9">depth</a> <font color="Maroon" title="c12">t</font>
 
<b>by </b><i><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="c13">d</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="E21:9"/><i><font color="Green" title="E14">A8</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">d</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="c12">t</font> &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="c13">d</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="c13">d</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="c7">v</font> )
 <b>by </b><i><a class="ref" href="msscyc_2.html#T2" target="_self" title="MSSCYC_2:th.2">Th2</a></i>;<br/>
<a NAME="E22:9"/>
not <font color="Maroon" title="c13">d</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="txt" href="#E18:9"><i><font color="Green" title="E13">A7</font></i></a>, <a class="txt" href="#E21:9"><i><font color="Green" title="E14">A8</font></i></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a></i>;<br/>
<b>hence </b><a NAME="E23:9"/>
contradiction
 <b>by </b><i><a class="txt" href="#E12:9"><i><font color="Green" title="E10">A4</font></i></a>, <a class="txt" href="#E18:9"><i><font color="Green" title="E13">A7</font></i></a>, <a class="txt" href="#E21:9"><i><font color="Green" title="E14">A8</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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