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<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="graph_1.html#NM3" title="GRAPH_1:NM.3">Graph</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">G</font> is  <a href="msscyc_1.html#V3" title="MSSCYC_1:attr.3">well-founded</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="msscyc_1.html#V2" title="MSSCYC_1:attr.2">directed_cycle-less</a>  )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:34_1_1_1"/>
( <font color="Maroon" title="c1">G</font> is  <a href="struct_0.html#V11" title="STRUCT_0:attr.11">void</a>  or  not <font color="Maroon" title="c1">G</font> is  <a href="struct_0.html#V11" title="STRUCT_0:attr.11">void</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:34_1_1_1_1"/>
<font color="Maroon" title="c1">G</font> is  <a href="struct_0.html#V11" title="STRUCT_0:attr.11">void</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">G</font> is  <a href="msscyc_1.html#V3" title="MSSCYC_1:attr.3">well-founded</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="msscyc_1.html#V2" title="MSSCYC_1:attr.2">directed_cycle-less</a>  )</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c2">G</font> = <font color="Maroon" title="c1">G</font> as   <a href="struct_0.html#V11" title="STRUCT_0:attr.11">void</a>  <a href="graph_1.html#NM3" title="GRAPH_1:NM.3">Graph</a> ;<br/><a NAME="E3:34_1_1_1_1"/>
<font color="Maroon" title="c2">G</font> is  <a href="msscyc_1.html#V2" title="MSSCYC_1:attr.2">directed_cycle-less</a> 
 ;<br/><span class="kw">hence </span><a NAME="E4:34_1_1_1_1"/>
( <font color="Maroon" title="c1">G</font> is  <a href="msscyc_1.html#V3" title="MSSCYC_1:attr.3">well-founded</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="msscyc_1.html#V2" title="MSSCYC_1:attr.2">directed_cycle-less</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:34_1_1_1_2"/>
 not <font color="Maroon" title="c1">G</font> is  <a href="struct_0.html#V11" title="STRUCT_0:attr.11">void</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">G</font> is  <a href="msscyc_1.html#V3" title="MSSCYC_1:attr.3">well-founded</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="msscyc_1.html#V2" title="MSSCYC_1:attr.2">directed_cycle-less</a>  )</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c2">G9</font> = <font color="Maroon" title="c1">G</font> as   non  <a href="struct_0.html#V11" title="STRUCT_0:attr.11">void</a>  <a href="graph_1.html#NM3" title="GRAPH_1:NM.3">Graph</a> ;<br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:34_1_1_1_2"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
<font color="Maroon" title="c1">G</font> is  <a href="msscyc_1.html#V3" title="MSSCYC_1:attr.3">well-founded</a> 
 <span class="kw">and </span><br/><a NAME="E4:34_1_1_1_2"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
 not <font color="Maroon" title="c1">G</font> is  <a href="msscyc_1.html#V2" title="MSSCYC_1:attr.2">directed_cycle-less</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">consider </span><font color="Maroon" title="c3">dc</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 <font color="Maroon" title="c2">G9</font><span class="kw"> such that </span><br/><a NAME="E6:34_1_1_1_2"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
 not <font color="Maroon" title="c3">dc</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 <span class="kw">and </span><br/><a NAME="E7:34_1_1_1_2"/><span class="lab"><font color="Green" title="E19">A4</font></span>: 
<font color="Maroon" title="c3">dc</font> is  <a href="msscyc_1.html#V1" title="MSSCYC_1:attr.1">cyclic</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_1_1_1_2"><span class="lab"><font color="Green" title="E17">A2</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c4">p</font> =  <a href="graph_2.html#K3" title="GRAPH_2:func.3">vertex-seq</a> <font color="Maroon" title="c3">dc</font>;<br/><a NAME="E8:34_1_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="graph_2.html#K3" title="GRAPH_2:func.3">vertex-seq</a> <font color="Maroon" title="c3">dc</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">dc</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:34_1_1_1_2"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="msscyc_1.html#T10" target="_self" onmouseover="rs('msscyc_1/T10')" onmouseout="rh()">Th9</a></span>;<br/><a NAME="E9:34_1_1_1_2"/><span class="kw">then </span>
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> <span class="p1">(<span class="default"><a href="graph_2.html#K3" title="GRAPH_2:func.3">vertex-seq</a> <font color="Maroon" title="c3">dc</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/><a NAME="E10:34_1_1_1_2"/><span class="kw">then </span>
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="graph_2.html#K3" title="GRAPH_2:func.3">vertex-seq</a> <font color="Maroon" title="c3">dc</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">v</font> = <span class="p1">(<span class="default"><a href="graph_2.html#K3" title="GRAPH_2:func.3">vertex-seq</a> <font color="Maroon" title="c3">dc</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 as    <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">G</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T11" onmouseover="rs('finseq_2/T11')" onmouseout="rh()">FINSEQ_2:11</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c6">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E13:34_1_1_1_2"/><span class="lab"><font color="Green" title="E20">A5</font></span>: 
 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 <font color="Maroon" title="c2">G9</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#K3" title="GRAPH_2:func.3">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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">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="c6">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_1_1_1_2"><span class="lab"><font color="Green" title="E16">A1</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c7">ch</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 <font color="Maroon" title="c2">G9</font><span class="kw"> such that </span><br/><a NAME="E15:34_1_1_1_2"/><span class="lab"><font color="Green" title="E21">A6</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">ch</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">and </span><br/><a NAME="E16:34_1_1_1_2"/><span class="lab"><font color="Green" title="E22">A7</font></span>: 
<font color="Maroon" title="c7">ch</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">dc</font> is   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <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 <font color="Maroon" title="c2">G9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:34_1_1_1_2"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E7:34_1_1_1_2"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="msscyc_1.html#T18" target="_self" onmouseover="rs('msscyc_1/T18')" onmouseout="rh()">Th17</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c8">ch</font> = <font color="Maroon" title="c7">ch</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <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 <font color="Maroon" title="c2">G9</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:34_1_1_1_2"><span class="lab"><font color="Green" title="E21">A6</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c9">cc</font> = <font color="Maroon" title="c8">ch</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">dc</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <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 <font color="Maroon" title="c2">G9</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:34_1_1_1_2"><span class="lab"><font color="Green" title="E22">A7</font></span></a></span>;<br/><a NAME="E19:34_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="graph_2.html#K3" title="GRAPH_2:func.3">vertex-seq</a> <font color="Maroon" title="c3">dc</font></span>)</span> <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">(<span class="default"><a href="graph_2.html#K3" title="GRAPH_2:func.3">vertex-seq</a> <font color="Maroon" title="c3">dc</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="c3">dc</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:34_1_1_1_2"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E7:34_1_1_1_2"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="msscyc_1.html#T17" target="_self" onmouseover="rs('msscyc_1/T17')" onmouseout="rh()">Th16</a></span>;<br/><a NAME="E20:34_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="graph_2.html#K3" title="GRAPH_2:func.3">vertex-seq</a> <font color="Maroon" title="c9">cc</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="c9">cc</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="c5">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:34_1_1_1_2"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="msscyc_1.html#T16" target="_self" onmouseover="rs('msscyc_1/T16')" onmouseout="rh()">Th15</a></span>;<br/><a NAME="E21:34_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A8</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">cc</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:34_1_1_1_2"><span class="lab"><font color="Green" title="E20">A5</font></span></a></span>;<br/><a NAME="E22:34_1_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">cc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">dc</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:34_1_1_1_2"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="ref" href="finseq_1.html#T22" onmouseover="rs('finseq_1/T22')" onmouseout="rh()">FINSEQ_1:22</a></span>;<br/><a NAME="E23:34_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 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="c9">cc</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/><span class="kw">hence </span><a NAME="E24:34_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:34_1_1_1_2"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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