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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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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#V3" title="GLIB_000:attr.3">loopless</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a>  )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:6_1_1"/><span class="lab"><font color="Green" title="E5">A1</font></span>: 
<font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">trivial</a> 
 <span class="kw">and </span><br/><a NAME="E2:6_1_1"/><span class="lab"><font color="Green" title="E6">A2</font></span>: 
<font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V3" title="GLIB_000:attr.3">loopless</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" title="c1">G</font> is  <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">given </span><font color="Maroon" title="c2">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><a NAME="E2:6_1_1_1"/><span class="lab"><font color="Green" title="E7">A3</font></span>: 
<font color="Maroon" title="c2">W</font> is  <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a> 
 ; <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="E3:6_1_1_1"/>
<font color="Maroon" title="c2">W</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:6_1_1_1"><span class="lab"><font color="Green" title="E7">A3</font></span></a>, <a class="ref" href="glib_001.html#T137" title="GLIB_001:th.137">GLIB_001:137</a></span>;<br/><a NAME="E4:6_1_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">W</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E5:6_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1_1"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E2:6_1_1"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="ref" href="glib_000.html#T26" title="GLIB_000:th.26">GLIB_000:26</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="E4:6_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E7">A4</font></span>: 
<font color="Maroon" title="c1">G</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_002.html#D2" target="_self" title="GLIB_002:def.2">Def2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c2">G9</font> = <font color="Maroon" title="c1">G</font> as   <a href="glib_000.html#V4" title="GLIB_000:attr.4">trivial</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1_1"><span class="lab"><font color="Green" title="E5">A1</font></span></a></span>;<br/>
<a NAME="E6:6_1_1"/>
<font color="Maroon" title="c2">G9</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> 
 
;<br/>
<span class="kw">hence </span><a NAME="E7:6_1_1"/>
<font color="Maroon" title="c1">G</font> is  <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:6_1_1"><span class="lab"><font color="Green" title="E7">A4</font></span></a>, <a class="ref" href="glib_002.html#D3" target="_self" title="GLIB_002:def.3">Def3</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/>


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