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<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_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="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  &amp; <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V6" title="GLIB_000:attr.6">non-Dmulti</a>  &amp; <font color="Maroon" title="c1">G</font> is  <a href="glib_012.html#V1" title="GLIB_012:attr.1">loopfull</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glpacy00.html#V4" title="GLPACY00:attr.4">Cycle-like</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:71_1_1"/><span class="lab"><font color="Green" title="E41">A1</font></span>: 
( <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  &amp; <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V6" title="GLIB_000:attr.6">non-Dmulti</a>  &amp; <font color="Maroon" title="c1">G</font> is  <a href="glib_012.html#V1" title="GLIB_012:attr.1">loopfull</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="glpacy00.html#V4" title="GLPACY00:attr.4">Cycle-like</a> </span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="71_1_1_1"><span class="kw">now </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">  for <font color="Olive" title="b1">v</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> holds  <font color="Olive" title="b1">v</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">v</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>; <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="c2">v</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2</span><br/><a NAME="E1:71_1_1_1"/><span class="lab"><font color="Green" title="E42">A2</font></span>: 
<font color="Maroon" title="c1">G</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.size()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="71_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E2:71_1_1_1_1"/><span class="lab"><font color="Green" title="E43">A3</font></span>: 
<font color="Maroon" title="c3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71_1_1"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="glib_012.html#D1" onmouseover="rs('glib_012/D1')" onmouseout="rh()">GLIB_012:def 1</a></span>;<br/>
<a NAME="E3:71_1_1_1_1"/><span class="lab"><font color="Green" title="E44">A4</font></span>: 
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71_1_1_1_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="ref" href="glib_000.html#D13" onmouseover="rs('glib_000/D13')" onmouseout="rh()">GLIB_000:def 13</a></span>;<br/>
<a NAME="E4:71_1_1_1_1"/>
 for <font color="Olive" title="b1">e1</font>, <font color="Olive" title="b2">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font> &amp; <font color="Olive" title="b2">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font> holds <br/><font color="Olive" title="b1">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">e2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71_1_1"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="zfmisc_1.html#D10" onmouseover="rs('zfmisc_1/D10')" onmouseout="rh()">ZFMISC_1:def 10</a></span>;<br/>
<a NAME="E5:71_1_1_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">e9</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">e9</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:71_1_1_1_1"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="ref" href="zfmisc_1.html#D10" onmouseover="rs('zfmisc_1/D10')" onmouseout="rh()">ZFMISC_1:def 10</a>, <a class="ref" href="zfmisc_1.html#T131" onmouseover="rs('zfmisc_1/T131')" onmouseout="rh()">ZFMISC_1:131</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:71_1_1_1_1"/>
<font color="Maroon" title="c1">G</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.size()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T30" onmouseover="rs('card_1/T30')" onmouseout="rh()">CARD_1:30</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><span class="kw">thus </span><font color="Maroon" title="c2">v</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  = 
1 <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71_1_1"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E1:71_1_1_1"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="ref" href="glib_000.html#T149" onmouseover="rs('glib_000/T149')" onmouseout="rh()">GLIB_000:149</a></span>
<br/>.= 
2

; <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>
<span class="kw">hence </span><a NAME="E3:71_1_1"/>
<font color="Maroon" title="c1">G</font> is  <a href="glpacy00.html#V4" title="GLPACY00:attr.4">Cycle-like</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71_1_1"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="glib_016.html#D4" onmouseover="rs('glib_016/D4')" onmouseout="rh()">GLIB_016:def 4</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/>


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