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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="graph_1.html#V5" title="GRAPH_1:attr.5">connected</a>   <a href="graph_1.html#V6" title="GRAPH_1:attr.6">finite</a>  <a href="graph_1.html#NM3" title="GRAPH_1:NM.3">Graph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">p</font> being  <a href="msscyc_1.html#V1" title="MSSCYC_1:attr.1">cyclic</a>  <a href="graph_1.html#NM6" title="GRAPH_1:NM.6">Path</a> of <font color="Maroon" title="c1">G</font> st <font color="Olive" title="b1">p</font> is  <a href="graph_3.html#V1" title="GRAPH_3:attr.1">Eulerian</a>  iff for <font color="Olive" title="b1">v</font> being  <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c1">G</font> holds   <a href="graph_1.html#K8" title="GRAPH_1:func.8">Degree</a> <font color="Olive" title="b1">v</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">E</font> = the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G</font>;<br/>
<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">v</font> being  <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c1">G</font> holds   <a href="graph_1.html#K8" title="GRAPH_1:func.8">Degree</a> <font color="Olive" title="b1">v</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  ) implies  ex <font color="Olive" title="b1">p</font> being  <a href="msscyc_1.html#V1" title="MSSCYC_1:attr.1">cyclic</a>  <a href="graph_1.html#NM6" title="GRAPH_1:NM.6">Path</a> of <font color="Maroon" title="c1">G</font> st <font color="Olive" title="b1">p</font> is  <a href="graph_3.html#V1" title="GRAPH_3:attr.1">Eulerian</a>  )</span>
<div class="add"><span class="kw">given </span><font color="Maroon" title="c3">c</font> being   <a href="msscyc_1.html#V1" title="MSSCYC_1:attr.1">cyclic</a>  <a href="graph_1.html#NM6" title="GRAPH_1:NM.6">Path</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><a NAME="E2:89_1"/><span class="lab"><font color="Green" title="E72">A1</font></span>: 
<font color="Maroon" title="c3">c</font> is  <a href="graph_3.html#V1" title="GRAPH_3:attr.1">Eulerian</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">v</font> being  <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c1">G</font> holds   <a href="graph_1.html#K8" title="GRAPH_1:func.8">Degree</a> <font color="Olive" title="b1">v</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">v</font> be   <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="graph_1.html#K8" title="GRAPH_1:func.8">Degree</a> <font color="Maroon" title="c4">v</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a> </span><br/><span class="kw">consider </span><font color="Maroon" title="c5">vs</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</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"> such that </span><br/><a NAME="E4:89_1"/><span class="lab"><font color="Green" title="E73">A2</font></span>: 
<font color="Maroon" title="c5">vs</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c3">c</font>
 <span class="kw">and </span><br/><a NAME="E5:89_1"/>
<font color="Maroon" title="c5">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">vs</font> <a href="funct_1.html#K1" title="FUNCT_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="c5">vs</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="msscyc_1.html#D2" title="MSSCYC_1:def.2">MSSCYC_1:def 2</a></span>;<br/><a NAME="E6:89_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">vs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 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="txt" href="#E2:89_1"><span class="lab"><font color="Green" title="E72">A1</font></span></a>, <a class="txt" href="#E4:89_1"><span class="lab"><font color="Green" title="E73">A2</font></span></a>, <a class="ref" href="graph_3.html#T63" target="_self" title="GRAPH_3:th.63">Th63</a></span>;<br/><a NAME="E7:89_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A3</font></span>: 
 <a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Maroon" title="c4">v</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">c</font></span>)</span> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:89_1"><span class="lab"><font color="Green" title="E73">A2</font></span></a>, <a class="txt" href="graph_3.html#E64"><span class="lab"><font color="Green" title="E52">Lm4</font></span></a></span>;<br/><a NAME="E8:89_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:89_1"><span class="lab"><font color="Green" title="E72">A1</font></span></a>, <a class="ref" href="graph_3.html#D14" target="_self" title="GRAPH_3:def.14">Def14</a></span>;<br/><span class="kw">hence </span><a NAME="E9:89_1"/>
 <a href="graph_1.html#K8" title="GRAPH_1:func.8">Degree</a> <font color="Maroon" title="c4">v</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:89_1"><span class="lab"><font color="Green" title="E74">A3</font></span></a>, <a class="ref" href="graph_3.html#T29" target="_self" title="GRAPH_3:th.29">Th29</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:89"/><span class="lab"><font color="Green" title="E72">A4</font></span>: 
for <font color="Olive" title="b1">v</font> being  <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c1">G</font> holds   <a href="graph_1.html#K8" title="GRAPH_1:func.8">Degree</a> <font color="Olive" title="b1">v</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p</font> being  <a href="msscyc_1.html#V1" title="MSSCYC_1:attr.1">cyclic</a>  <a href="graph_1.html#NM6" title="GRAPH_1:NM.6">Path</a> of <font color="Maroon" title="c1">G</font> st <font color="Olive" title="b1">p</font> is  <a href="graph_3.html#V1" title="GRAPH_3:attr.1">Eulerian</a> </span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_2"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:89_2"/>
( <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="89_2_1"><span class="kw">suppose </span></a><a NAME="E1:89_2_1"/><span class="lab"><font color="Green" title="E73">A5</font></span>: 
<font color="Maroon" title="c1">G</font> is  <a href="struct_0.html#V11" title="STRUCT_0:attr.11">void</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p</font> being  <a href="msscyc_1.html#V1" title="MSSCYC_1:attr.1">cyclic</a>  <a href="graph_1.html#NM6" title="GRAPH_1:NM.6">Path</a> of <font color="Maroon" title="c1">G</font> st <font color="Olive" title="b1">p</font> is  <a href="graph_3.html#V1" title="GRAPH_3:attr.1">Eulerian</a> </span><br/><div class="add"><a NAME="E2:89_2_1"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">G</font> <a href="graph_3.html#K9" title="GRAPH_3:func.9">-CycleSet</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_3.html#T56" target="_self" title="GRAPH_3:th.56">Th56</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">ec</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as   <a href="msscyc_1.html#V1" title="MSSCYC_1:attr.1">cyclic</a>  <a href="graph_1.html#NM6" title="GRAPH_1:NM.6">Path</a> of <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_3.html#D8" target="_self" title="GRAPH_3:def.8">Def8</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c3">ec</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">ec</font> is  <a href="graph_3.html#V1" title="GRAPH_3:attr.1">Eulerian</a> </span><br/><a NAME="E4:89_2_1"/>
the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:89_2_1"><span class="lab"><font color="Green" title="E73">A5</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:89_2_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">ec</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G</font>
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="graph_3.html#D14" target="_self" title="GRAPH_3:def.14">GRAPH_3:def 14</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="89_2_2"><span class="kw">suppose </span></a><a NAME="E1:89_2_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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p</font> being  <a href="msscyc_1.html#V1" title="MSSCYC_1:attr.1">cyclic</a>  <a href="graph_1.html#NM6" title="GRAPH_1:NM.6">Path</a> of <font color="Maroon" title="c1">G</font> st <font color="Olive" title="b1">p</font> is  <a href="graph_3.html#V1" title="GRAPH_3:attr.1">Eulerian</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">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#V5" title="GRAPH_1:attr.5">connected</a>   <a href="graph_1.html#V6" title="GRAPH_1:attr.6">finite</a>  <a href="graph_1.html#NM3" title="GRAPH_1:NM.3">Graph</a> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c4">V</font> =  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> as   <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c3">G9</font> ;<br/><span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>,   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c3">G9</font> <a href="graph_3.html#K9" title="GRAPH_3:func.9">-CycleSet</a> ) <span class="kw">-&gt; </span>   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c3">G9</font> <a href="graph_3.html#K9" title="GRAPH_3:func.9">-CycleSet</a>  =  <a href="graph_3.html#K14" title="GRAPH_3:func.14">ExtendCycle</a> $2;<br/><a NAME="E4:89_2_2"/>
 <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G</font> <a href="graph_3.html#K13" title="GRAPH_3:func.13">-CycleSet</a> <font color="Maroon" title="c4">V</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G</font> <a href="graph_3.html#K13" title="GRAPH_3:func.13">-CycleSet</a> <font color="Maroon" title="c4">V</font>
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">ec</font> =  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G</font> <a href="graph_3.html#K13" title="GRAPH_3:func.13">-CycleSet</a> <font color="Maroon" title="c4">V</font></span>)</span> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c3">G9</font> <a href="graph_3.html#K9" title="GRAPH_3:func.9">-CycleSet</a>  ;<br/><span class="kw">consider </span><font color="Maroon" title="c6">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c3">G9</font> <a href="graph_3.html#K9" title="GRAPH_3:func.9">-CycleSet</a> </span>)</span><span class="kw"> such that </span><br/><a NAME="E7:89_2_2"/><span class="lab"><font color="Green" title="E73">A6</font></span>: 
( <font color="Maroon" title="c6">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">ec</font> &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c6">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <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">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c6">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font>) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S12" title="NAT_1:sch.12">NAT_1:sch 12</a>();<br/></span><div><span class="lab"><font color="Green" title="E74">A7</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="89_2_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">v</font> be   <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Maroon" title="c7">v</font>,the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a> </span><br/><a NAME="E1:89_2_2_1"/>
 <a href="graph_1.html#K8" title="GRAPH_1:func.8">Degree</a> <font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Maroon" title="c7">v</font>,the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_3.html#T29" target="_self" title="GRAPH_3:th.29">Th29</a></span>;<br/><span class="kw">hence </span><a NAME="E2:89_2_2_1"/>
 <a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Maroon" title="c7">v</font>,the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:89"><span class="lab"><font color="Green" title="E72">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E9:89_2_2"/>
 <a href="graph_1.html#K8" title="GRAPH_1:func.8">Degree</a> <font color="Maroon" title="c4">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Maroon" title="c4">V</font>,the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_3.html#T29" target="_self" title="GRAPH_3:th.29">Th29</a></span>;<br/><a NAME="E10:89_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A8</font></span>: 
 <a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Maroon" title="c4">V</font>,the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_3.html#T38" target="_self" title="GRAPH_3:th.38">Th38</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_2_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <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> ] <span class="kw">means </span> ex <font color="Olive" title="b1">c</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c3">G9</font> <a href="graph_3.html#K9" title="GRAPH_3:func.9">-CycleSet</a>  st <br/>( not <font color="Olive" title="b1">c</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> $1 &amp; $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">c</font></span>)</span> );<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">E</font> = the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G</font> as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_1.html#D9" title="GRAPH_1:def.9">GRAPH_1:def 9</a></span>;<br/><span class="kw">assume </span><a NAME="E2:89_2_2_2"/><span class="lab"><font color="Green" title="E76">A9</font></span>: 
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> <br/> for <font color="Olive" title="b2">c</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c3">G9</font> <a href="graph_3.html#K9" title="GRAPH_3:func.9">-CycleSet</a>   holds <br/>(  not <font color="Olive" title="b2">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> or  not  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b2">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E3:89_2_2_2"/><span class="lab"><font color="Green" title="E77">A10</font></span>: 
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>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_2_2_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">n</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c9">c</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c3">G9</font> <a href="graph_3.html#K9" title="GRAPH_3:func.9">-CycleSet</a> <span class="kw"> such that </span><a NAME="E2:89_2_2_2_1"/><span class="lab"><font color="Green" title="E78">A11</font></span>: 
not <font color="Maroon" title="c9">c</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 <span class="kw">and </span><br/><a NAME="E3:89_2_2_2_1"/><span class="lab"><font color="Green" title="E79">A12</font></span>: 
<font color="Maroon" title="c9">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">n</font>
 <span class="kw">and </span><br/><a NAME="E4:89_2_2_2_1"/><span class="lab"><font color="Green" title="E80">A13</font></span>: 
<font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">c</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c10">r</font> =  <a href="graph_3.html#K14" title="GRAPH_3:func.14">ExtendCycle</a> <font color="Maroon" title="c9">c</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c3">G9</font> <a href="graph_3.html#K9" title="GRAPH_3:func.9">-CycleSet</a>  ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c10">r</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( not <font color="Maroon" title="c10">r</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; <font color="Maroon" title="c10">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">r</font></span>)</span> )</span><br/>

<a NAME="E6:89_2_2_2_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:89_2_2_2"><span class="lab"><font color="Green" title="E76">A9</font></span></a>, <a class="txt" href="#E3:89_2_2_2_1"><span class="lab"><font color="Green" title="E79">A12</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:89_2_2_2_1"/>
not <font color="Maroon" title="c10">r</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:89"><span class="lab"><font color="Green" title="E72">A4</font></span></a>, <a class="txt" href="#E2:89_2_2_2_1"><span class="lab"><font color="Green" title="E78">A11</font></span></a>, <a class="ref" href="graph_3.html#T62" target="_self" title="GRAPH_3:th.62">Th62</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">r</font></span>)</span> )</span><br/>

<span class="kw">thus </span><a NAME="E8:89_2_2_2_1"/>
<font color="Maroon" title="c10">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:89_2_2"><span class="lab"><font color="Green" title="E73">A6</font></span></a>, <a class="txt" href="#E3:89_2_2_2_1"><span class="lab"><font color="Green" title="E79">A12</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">r</font></span>)</span></span><br/>

<a NAME="E9:89_2_2_2_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:89_2_2_2"><span class="lab"><font color="Green" title="E76">A9</font></span></a>, <a class="txt" href="#E3:89_2_2_2_1"><span class="lab"><font color="Green" title="E79">A12</font></span></a></span>;<br/>
<a NAME="E10:89_2_2_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">r</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:89"><span class="lab"><font color="Green" title="E72">A4</font></span></a>, <a class="txt" href="#E2:89_2_2_2_1"><span class="lab"><font color="Green" title="E78">A11</font></span></a>, <a class="txt" href="#E4:89_2_2_2_1"><span class="lab"><font color="Green" title="E80">A13</font></span></a>, <a class="ref" href="graph_3.html#T62" target="_self" title="GRAPH_3:th.62">Th62</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:89_2_2_2_1"/>
<font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">r</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E4:89_2_2_2"/><span class="lab"><font color="Green" title="E78">A14</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_2_2_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span>
<font color="Maroon" title="c5">ec</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( not <font color="Maroon" title="c5">ec</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; <font color="Maroon" title="c5">ec</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">ec</font></span>)</span> )</span><br/>

<span class="kw">thus </span><a NAME="E1:89_2_2_2_2"/>
not <font color="Maroon" title="c5">ec</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:89_2_2"><span class="lab"><font color="Green" title="E74">A7</font></span></a>, <a class="txt" href="#E10:89_2_2"><span class="lab"><font color="Green" title="E75">A8</font></span></a>, <a class="ref" href="graph_3.html#T60" target="_self" title="GRAPH_3:th.60">Th60</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">ec</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">ec</font></span>)</span> )</span><br/>

<span class="kw">thus </span><a NAME="E2:89_2_2_2_2"/>
<font color="Maroon" title="c5">ec</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:89_2_2"><span class="lab"><font color="Green" title="E73">A6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">ec</font></span>)</span></span><br/>

<span class="kw">thus </span><a NAME="E3:89_2_2_2_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">ec</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E5:89_2_2_2"/>
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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E4:89_2_2_2"><span class="lab"><font color="Green" title="E78">A14</font></span></a>, <a class="txt" href="#E3:89_2_2_2"><span class="lab"><font color="Green" title="E77">A10</font></span></a>);<br/></span><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">c</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">G</font> <a href="graph_3.html#K9" title="GRAPH_3:func.9">-CycleSet</a> <span class="kw"> such that </span><br/><a NAME="E7:89_2_2_2"/>
not <font color="Maroon" title="c8">c</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 <span class="kw">and </span><br/><a NAME="E8:89_2_2_2"/>
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">E</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E9:89_2_2_2"/><span class="lab"><font color="Green" title="E79">A15</font></span>: 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">E</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">c</font></span>)</span>
 ;<br/><a NAME="E10:89_2_2_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">c</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/><a NAME="E11:89_2_2_2"/><span class="kw">then </span>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">c</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T44" title="NAT_1:th.44">NAT_1:44</a></span>;<br/><a NAME="E12:89_2_2_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">E</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">E</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:89_2_2_2"><span class="lab"><font color="Green" title="E79">A15</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E13:89_2_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">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> , <font color="Maroon" title="c8">c</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">G</font> <a href="graph_3.html#K9" title="GRAPH_3:func.9">-CycleSet</a> <span class="kw"> such that </span><br/><a NAME="E13:89_2_2"/>
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">n</font>
 <span class="kw">and </span><br/><a NAME="E14:89_2_2"/><span class="lab"><font color="Green" title="E76">A16</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G</font>
 ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c9">c</font> = <font color="Maroon" title="c8">c</font> as   <a href="msscyc_1.html#V1" title="MSSCYC_1:attr.1">cyclic</a>  <a href="graph_1.html#NM6" title="GRAPH_1:NM.6">Path</a> of <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_3.html#D8" target="_self" title="GRAPH_3:def.8">Def8</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c9">c</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">c</font> is  <a href="graph_3.html#V1" title="GRAPH_3:attr.1">Eulerian</a> </span><br/><span class="kw">thus </span><a NAME="E16:89_2_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:89_2_2"><span class="lab"><font color="Green" title="E76">A16</font></span></a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="graph_3.html#D14" target="_self" title="GRAPH_3:def.14">GRAPH_3:def 14</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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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