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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>for <font color="Olive" title="b1">G</font> being  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a>   <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>  st <font color="Olive" title="b1">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> $1 &amp; <font color="Olive" title="b1">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="Olive" title="b1">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 holds <br/><font color="Olive" title="b1">G</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> ;<br/>
<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="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>  iff ( <font color="Maroon" title="c1">G</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</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/>

<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"> ( <font color="Maroon" title="c1">G</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</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>
<div class="add"><span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>for <font color="Olive" title="b1">T</font> being  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a>   <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>  st <font color="Olive" title="b1">T</font> <a href="glib_000.html#K15" title="GLIB_000:func.15">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/>$1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">T</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;<br/><span class="kw">assume </span><a NAME="E1:78_1"/><span class="lab"><font color="Green" title="E47">A1</font></span>: 
<font color="Maroon" title="c1">G</font> is  <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</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#V2" title="GLIB_002:attr.2">acyclic</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">hence </span><a NAME="E2:78_1"/>
<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="Red">::  thesis: </font></span></a><span class="hide"> <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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">T</font> be   <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a>   <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">T</font> <a href="glib_000.html#K15" title="GLIB_000:func.15">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">T</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">set </span><font color="Maroon" title="c3">VT</font> =  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">T</font>;<br/><span class="kw">set </span><font color="Maroon" title="c4">ET</font> =  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">T</font>;<br/><span class="kw">assume </span><a NAME="E1:78_1_1"/>
<font color="Maroon" title="c2">T</font> <a href="glib_000.html#K15" title="GLIB_000:func.15">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">T</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/><a NAME="E2:78_1_1"/><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="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D26" title="GLIB_000:def.26">GLIB_000:def 26</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">v</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:78_1_1"/><span class="lab"><font color="Green" title="E48">A2</font></span>: 
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">T</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="Maroon" title="c5">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_2.html#T60" title="CARD_2:th.60">CARD_2:60</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c6">v</font> = <font color="Maroon" title="c5">v</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">T</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:78_1_1"><span class="lab"><font color="Green" title="E48">A2</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:78_1_1_1"/>
 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>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">T</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/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:78_1_1_1"/><span class="lab"><font color="Green" title="E49">A3</font></span>: 
<font color="Maroon" title="c7">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="c2">T</font>
 ;<br/><a NAME="E4:78_1_1_1"/>
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c2">T</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:78_1_1"><span class="lab"><font color="Green" title="E48">A2</font></span></a>, <a class="txt" href="#E3:78_1_1_1"><span class="lab"><font color="Green" title="E49">A3</font></span></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></span>;<br/><a NAME="E5:78_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A4</font></span>: 
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c2">T</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E6:78_1_1_1"/>
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c2">T</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:78_1_1"><span class="lab"><font color="Green" title="E48">A2</font></span></a>, <a class="txt" href="#E3:78_1_1_1"><span class="lab"><font color="Green" title="E49">A3</font></span></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></span>;<br/><a NAME="E7:78_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c2">T</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E8:78_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:78_1_1_1"><span class="lab"><font color="Green" title="E49">A3</font></span></a>, <a class="txt" href="#E5:78_1_1_1"><span class="lab"><font color="Green" title="E50">A4</font></span></a>, <a class="ref" href="glib_000.html#D15" title="GLIB_000:def.15">GLIB_000:def 15</a></span>;<br/><a NAME="E9:78_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">T</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c7">e</font>,<font color="Maroon" title="c6">v</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_001.html#T157" title="GLIB_001:th.157">GLIB_001:157</a></span>;<br/><span class="kw">hence </span><a NAME="E10:78_1_1_1"/>
contradiction
 <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>; <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="E7:78_1_1"/><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="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/><a NAME="E8:78_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">T</font> <a href="glib_000.html#K17" title="GLIB_000:func.17">.size()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D27" title="GLIB_000:def.27">GLIB_000:def 27</a></span>;<br/><span class="kw">hence </span><a NAME="E9:78_1_1"/>
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">T</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="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E4:78_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A5</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[1]
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">k</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</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">T</font> being  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a>   <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>  st <font color="Olive" title="b1">T</font> <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="c2">k</font> holds <br/><font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">T</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 for <font color="Olive" title="b1">T</font> being  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a>   <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>  st <font color="Olive" title="b1">T</font> <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="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 holds <br/><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">T</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">assume </span><a NAME="E1:78_1_2"/><span class="lab"><font color="Green" title="E49">A6</font></span>: 
for <font color="Olive" title="b1">T</font> being  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a>   <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>  st <font color="Olive" title="b1">T</font> <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="c2">k</font> holds <br/><font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">T</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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">T</font> being  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a>   <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>  st <font color="Olive" title="b1">T</font> <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="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 holds <br/><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">T</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">let </span><font color="Maroon" title="c3">T</font> be   <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a>   <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">T</font> <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="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies <font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">T</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">assume </span><a NAME="E2:78_1_2"/><span class="lab"><font color="Green" title="E50">A7</font></span>: 
<font color="Maroon" title="c3">T</font> <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="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <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="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">T</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/><a NAME="E3:78_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">T</font> <a href="glib_000.html#K15" title="GLIB_000:func.15">.order()</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T3" title="XCMPLX_1:th.3">XCMPLX_1:3</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">aT</font> = <font color="Maroon" title="c3">T</font> as   <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a>  non  <a href="glib_000.html#V4" title="GLIB_000:attr.4">trivial</a>   <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T29" title="GLIB_000:th.29">GLIB_000:29</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c5">v</font> being   <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c4">aT</font>;<br/><span class="kw">consider </span><font color="Maroon" title="c6">T2</font> being   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c4">aT</font>,<font color="Maroon" title="c5">v</font>;<br/><a NAME="E7:78_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">T2</font> <a href="glib_000.html#K15" title="GLIB_000:func.15">.order()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:78_1_2"><span class="lab"><font color="Green" title="E50">A7</font></span></a>, <a class="ref" href="glib_000.html#T51" title="GLIB_000:th.51">GLIB_000:51</a></span>;<br/><a NAME="E8:78_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A8</font></span>: 
<font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">T2</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="txt" href="#E1:78_1_2"><span class="lab"><font color="Green" title="E49">A6</font></span></a></span>;<br/> <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a> </span>)</span> = 
<font color="Maroon" title="c5">v</font> <a href="glib_000.html#K35" title="GLIB_000:func.35">.degree()</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T22" title="GLIB_000:th.22">GLIB_000:22</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D54" title="GLIB_000:def.54">GLIB_000:def 54</a></span>
;<br/><span class="kw">hence </span><a NAME="E10:78_1_2"/>
<font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">T</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="txt" href="#E8:78_1_2"><span class="lab"><font color="Green" title="E51">A8</font></span></a>, <a class="ref" href="glib_000.html#T51" title="GLIB_000:th.51">GLIB_000:51</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="E6:78_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A9</font></span>: 
for <font color="Olive" title="b1">k</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ;<br/><a NAME="E7:78_1"/>
for <font color="Olive" title="b1">k</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S10" title="NAT_1:sch.10">NAT_1:sch 10</a>(<a class="txt" href="#E4:78_1"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="txt" href="#E6:78_1"><span class="lab"><font color="Green" title="E49">A9</font></span></a>);<br/></span><span class="kw">hence </span><a NAME="E8:78_1"/>
<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="txt" href="#E1:78_1"><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="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:78"/><span class="lab"><font color="Green" title="E47">A10</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">and </span><br/><a NAME="E3:78"/><span class="lab"><font color="Green" title="E48">A11</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="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="78_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">k</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <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"> ( ( for <font color="Olive" title="b1">G</font> being  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a>   <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>  st <font color="Olive" title="b1">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> <font color="Maroon" title="c2">k</font> &amp; <font color="Olive" title="b1">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="Olive" title="b1">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 holds <br/><font color="Olive" title="b1">G</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>  ) implies for <font color="Olive" title="b1">G</font> being  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a>   <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>  st <font color="Olive" title="b1">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> <font color="Maroon" title="c2">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b1">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="Olive" title="b1">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 holds <br/><font color="Olive" title="b1">G</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:78_2"/><span class="lab"><font color="Green" title="E49">A12</font></span>: 
for <font color="Olive" title="b1">G</font> being  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a>   <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>  st <font color="Olive" title="b1">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> <font color="Maroon" title="c2">k</font> &amp; <font color="Olive" title="b1">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="Olive" title="b1">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 holds <br/><font color="Olive" title="b1">G</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</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">G</font> being  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a>   <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>  st <font color="Olive" title="b1">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> <font color="Maroon" title="c2">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b1">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="Olive" title="b1">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 holds <br/><font color="Olive" title="b1">G</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">G</font> be   <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a>   <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">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> <font color="Maroon" title="c2">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c3">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="c3">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="c3">G</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>  )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:78_2"/><span class="lab"><font color="Green" title="E50">A13</font></span>: 
<font color="Maroon" title="c3">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> <font color="Maroon" title="c2">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">and </span><br/><a NAME="E3:78_2"/><span class="lab"><font color="Green" title="E51">A14</font></span>: 
<font color="Maroon" title="c3">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="c3">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="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">G</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> </span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:78_2_1"/>
<font color="Maroon" title="c3">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> 1
 ; <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="E2:78_2_1"/><span class="kw">then </span>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="c2">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:78_2"><span class="lab"><font color="Green" title="E50">A13</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:78_2_1"/>
contradiction
 ; <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">reconsider </span><font color="Maroon" title="c4">aG</font> = <font color="Maroon" title="c3">G</font> as   <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a>  non  <a href="glib_000.html#V4" title="GLIB_000:attr.4">trivial</a>   <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T29" title="GLIB_000:th.29">GLIB_000:29</a></span>;<br/><a NAME="E6:78_2"/>
 <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c3">G</font> <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="#E2:78_2"><span class="lab"><font color="Green" title="E50">A13</font></span></a>, <a class="txt" href="#E3:78_2"><span class="lab"><font color="Green" title="E51">A14</font></span></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a>, <a class="ref" href="glib_000.html#D27" title="GLIB_000:def.27">GLIB_000:def 27</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">v</font>, <font color="Maroon" title="c6">v2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c4">aG</font><span class="kw"> such that </span><br/><a NAME="E8:78_2"/>
<font color="Maroon" title="c5">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">v2</font>
 <span class="kw">and </span><br/><a NAME="E9:78_2"/><span class="lab"><font color="Green" title="E52">A15</font></span>: 
<font color="Maroon" title="c5">v</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a> 
 <span class="kw">and </span><br/><a NAME="E10:78_2"/>
<font color="Maroon" title="c6">v2</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a> 
 <span class="kw">and </span><br/><a NAME="E11:78_2"/>
<font color="Maroon" title="c6">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">aG</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c5">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="glib_002.html#E35"><span class="lab"><font color="Green" title="E34">Lm23</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c7">G2</font> being   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c5">v</font>;<br/><a NAME="E13:78_2"/><span class="lab"><font color="Green" title="E53">A16</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c7">G2</font> <a href="glib_000.html#K15" title="GLIB_000:func.15">.order()</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="c4">aG</font> <a href="glib_000.html#K15" title="GLIB_000:func.15">.order()</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">G2</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> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">aG</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#T51" title="GLIB_000:th.51">GLIB_000:51</a></span>;<br/> <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a> </span>)</span> = 
<font color="Maroon" title="c5">v</font> <a href="glib_000.html#K35" title="GLIB_000:func.35">.degree()</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T22" title="GLIB_000:th.22">GLIB_000:22</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:78_2"><span class="lab"><font color="Green" title="E52">A15</font></span></a>, <a class="ref" href="glib_000.html#D54" title="GLIB_000:def.54">GLIB_000:def 54</a></span>
;<br/><a NAME="E15:78_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A17</font></span>: 
<font color="Maroon" title="c7">G2</font> 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="#E1:78_2"><span class="lab"><font color="Green" title="E49">A12</font></span></a>, <a class="txt" href="#E2:78_2"><span class="lab"><font color="Green" title="E50">A13</font></span></a>, <a class="txt" href="#E3:78_2"><span class="lab"><font color="Green" title="E51">A14</font></span></a>, <a class="txt" href="#E13:78_2"><span class="lab"><font color="Green" title="E53">A16</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c8">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E17:78_2"/><span class="lab"><font color="Green" title="E55">A18</font></span>: 
<font color="Maroon" title="c5">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a>  <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="Maroon" title="c8">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">and </span><br/><a NAME="E18:78_2"/><span class="lab"><font color="Green" title="E56">A19</font></span>: 
not <font color="Maroon" title="c8">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c3">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:78_2"><span class="lab"><font color="Green" title="E52">A15</font></span></a>, <a class="ref" href="glib_000.html#D53" title="GLIB_000:def.53">GLIB_000:def 53</a></span>;<br/><a NAME="E19:78_2"/>
<font color="Maroon" title="c8">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:78_2"><span class="lab"><font color="Green" title="E55">A18</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E20:78_2"/>
<font color="Maroon" title="c3">G</font> 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="#E15:78_2"><span class="lab"><font color="Green" title="E54">A17</font></span></a>, <a class="txt" href="#E18:78_2"><span class="lab"><font color="Green" title="E56">A19</font></span></a>, <a class="ref" href="glib_002.html#T3" target="_self" title="GLIB_002:th.3">Th3</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="E5:78"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A20</font></span>: 
for <font color="Olive" title="b1">k</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">G</font> be   <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a>   <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">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> 1 &amp; <font color="Maroon" title="c2">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="c2">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="c2">G</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>  )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:78_3"/><span class="lab"><font color="Green" title="E50">A21</font></span>: 
<font color="Maroon" title="c2">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> 1
 <span class="kw">and </span><br/><a NAME="E2:78_3"/>
<font color="Maroon" title="c2">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="c2">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="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">G</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> </span><br/><span class="kw">consider </span><font color="Maroon" title="c3">v</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G</font><span class="kw"> such that </span><br/><a NAME="E4:78_3"/><span class="lab"><font color="Green" title="E51">A22</font></span>: 
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78_3"><span class="lab"><font color="Green" title="E50">A21</font></span></a>, <a class="ref" href="glib_000.html#T30" title="GLIB_000:th.30">GLIB_000:30</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">v1</font>, <font color="Maroon" title="c5">v2</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G</font>; <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">W</font> being   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c2">G</font> st <font color="Olive" title="b1">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c5">v2</font></span><br/><a NAME="E1:78_3_1"/>
( <font color="Maroon" title="c4">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> &amp; <font color="Maroon" title="c5">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:78_3"><span class="lab"><font color="Green" title="E51">A22</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E2:78_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">G</font> <a href="glib_001.html#K1" title="GLIB_001:func.1">.walkOf</a> <font color="Maroon" title="c3">v</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c5">v2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T14" title="GLIB_001:th.14">GLIB_001:14</a></span>;<br/><span class="kw">hence </span><a NAME="E3:78_3_1"/>
 ex <font color="Olive" title="b1">W</font> being   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c2">G</font> st <font color="Olive" title="b1">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c5">v2</font>
 ; <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">hence </span><a NAME="E6:78_3"/>
<font color="Maroon" title="c2">G</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_002.html#D1" target="_self" title="GLIB_002:def.1">Def1</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="E7:78"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A23</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
;<br/>
<a NAME="E8:78"/>
for <font color="Olive" title="b1">k</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S10" title="NAT_1:sch.10">NAT_1:sch 10</a>(<a class="txt" href="#E7:78"><span class="lab"><font color="Green" title="E50">A23</font></span></a>, <a class="txt" href="#E5:78"><span class="lab"><font color="Green" title="E49">A20</font></span></a>);<br/></span>
<a NAME="E9:78"/><span class="kw">then </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">by</span> <span class="lab"><a class="txt" href="#E2:78"><span class="lab"><font color="Green" title="E47">A10</font></span></a>, <a class="txt" href="#E3:78"><span class="lab"><font color="Green" title="E48">A11</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:78"/>
<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="#E2:78"><span class="lab"><font color="Green" title="E47">A10</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>
