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<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a>  <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="firebrick">::  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>  iff ( <font color="Maroon" title="c1">G</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>  &amp; <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K15" title="GLIB_000:func.15">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K17" title="GLIB_000:func.17">.size()</a></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )</span><br/>

<span class="kw">thus </span><a NAME="E1:79"/>
( <font color="Maroon" title="c1">G</font> is  <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a>  implies ( <font color="Maroon" title="c1">G</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>  &amp; <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K15" title="GLIB_000:func.15">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K17" title="GLIB_000:func.17">.size()</a></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_002.html#T46" target="_self" title="GLIB_002:th.46">Th46</a></span>; <a class="txt" 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_002.html#V1" title="GLIB_002:attr.1">connected</a>  &amp; <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K15" title="GLIB_000:func.15">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K17" title="GLIB_000:func.17">.size()</a></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 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="E2:79"/><span class="lab"><font color="Green" title="E47">A1</font></span>: 
<font color="Maroon" title="c1">G</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> 
 <span class="kw">and </span><br/><a NAME="E3:79"/><span class="lab"><font color="Green" title="E48">A2</font></span>: 
<font color="Maroon" title="c1">G</font> <a href="glib_000.html#K15" title="GLIB_000:func.15">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K17" title="GLIB_000:func.17">.size()</a></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" 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_002.html#V3" title="GLIB_002:attr.3">Tree-like</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:79_1"/>
 not <font color="Maroon" title="c1">G</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </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><br/><a NAME="E3:79_1"/><span class="lab"><font color="Green" title="E49">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> 
 <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">set </span><font color="Maroon" title="c3">e</font> =  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a></span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c4">G2</font> =  the   <a href="glib_000.html#NM8" title="GLIB_000:NM.8">removeEdge</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a></span>)</span></span>)</span>;<br/><a NAME="E4:79_1"/><span class="lab"><font color="Green" title="E50">A4</font></span>: 
<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="#E3:79_1"><span class="lab"><font color="Green" title="E49">A3</font></span></a>, <a class="ref" href="glib_001.html#T136" title="GLIB_001:th.136">GLIB_001:136</a></span>;<br/><a NAME="E5:79_1"/><span class="kw">then </span>
 <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a></span>)</span> <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> 
 ;<br/><a NAME="E6:79_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A5</font></span>: 
(  the   <a href="glib_000.html#NM8" title="GLIB_000:NM.8">removeEdge</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a></span>)</span></span>)</span> <a href="glib_000.html#K15" title="GLIB_000:func.15">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K15" title="GLIB_000:func.15">.order()</a>  &amp; <span class="p1">(<span class="default"> the   <a href="glib_000.html#NM8" title="GLIB_000:NM.8">removeEdge</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a></span>)</span></span>)</span> <a href="glib_000.html#K17" title="GLIB_000:func.17">.size()</a></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K17" title="GLIB_000:func.17">.size()</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T52" title="GLIB_000:th.52">GLIB_000:52</a></span>;<br/><a NAME="E7:79_1"/>
 the   <a href="glib_000.html#NM8" title="GLIB_000:NM.8">removeEdge</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a></span>)</span></span>)</span> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:79"><span class="lab"><font color="Green" title="E47">A1</font></span></a>, <a class="txt" href="#E3:79_1"><span class="lab"><font color="Green" title="E49">A3</font></span></a>, <a class="txt" href="#E4:79_1"><span class="lab"><font color="Green" title="E50">A4</font></span></a>, <a class="ref" href="glib_002.html#T5" target="_self" title="GLIB_002:th.5">Th5</a></span>;<br/><a NAME="E8:79_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"> the   <a href="glib_000.html#NM8" title="GLIB_000:NM.8">removeEdge</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a></span>)</span></span>)</span> <a href="glib_000.html#K17" title="GLIB_000:func.17">.size()</a></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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"><span class="p2">(<span class="default"> the   <a href="glib_000.html#NM8" title="GLIB_000:NM.8">removeEdge</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a></span>)</span></span>)</span> <a href="glib_000.html#K17" title="GLIB_000:func.17">.size()</a></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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="#E3:79"><span class="lab"><font color="Green" title="E48">A2</font></span></a>, <a class="txt" href="#E6:79_1"><span class="lab"><font color="Green" title="E51">A5</font></span></a>, <a class="ref" href="glib_002.html#T40" target="_self" title="GLIB_002:th.40">Th40</a></span>;<br/><span class="kw">hence </span><a NAME="E9:79_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>; <a class="txt" 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="E5:79"/>
<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="#E2:79"><span class="lab"><font color="Green" title="E47">A1</font></span></a></span>; <a class="txt" 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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