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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="glib_000.html#V2">finite</a> <a href="glib_000.html#NM2">_Graph</a>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:79_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="glib_002.html#V3" target="_self">Tree-like</a>
 ;<br/><b>hence </b><a NAME="E2:79_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="glib_002.html#V2" target="_self">acyclic</a>
 <b>by </b><i><a class="ref" href="glib_002.html#D3" target="_self">Def3</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means for <font color="Olive">b<sub>1</sub></font> being <a href="glib_000.html#V2">finite</a> <a href="glib_002.html#V3" target="_self">Tree-like</a> <a href="glib_000.html#NM2">_Graph</a> st <font color="Olive">b<sub>1</sub></font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="glib_000.html#K18">.size()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="glib_000.html#V2">finite</a> <a href="glib_002.html#V3" target="_self">Tree-like</a> <a href="glib_000.html#NM2">_Graph</a>;<br/><b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">c<sub>2</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">c<sub>2</sub></font>;<br/><b>assume </b><a NAME="E1:79_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> 1
 ;<br/><a NAME="E2:79_1_1"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="glib_000.html#D26">GLIB_000:def 26</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:79_1_1"/><i><font color="Green">E49</font></i>: 
 <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="card_2.html#T60">CARD_2:60</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="glib_002.html#E4:79_1_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:79_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:79_1_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E4:79_1_1_1"/>
( <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="txt" href="glib_002.html#E4:79_1_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="glib_002.html#E3:79_1_1_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><a NAME="E5:79_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E6:79_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E3:79_1_1_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="glib_000.html#D15">GLIB_000:def 15</a></i>;<br/><a NAME="E7:79_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font> is <a href="glib_001.html#V8">Cycle-like</a>
 <b>by </b><i><a class="ref" href="glib_001.html#T157">GLIB_001:157</a></i>;<br/><b>hence </b><a NAME="E8:79_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="glib_002.html#D2" target="_self">Def2</a></i>;<br/></div><b>end;</b></div><a NAME="E7:79_1_1"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="card_1.html#T78">CARD_1:78</a>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E8:79_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="glib_000.html#K18">.size()</a>  <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="glib_000.html#D27">GLIB_000:def 27</a></i>;<br/><b>hence </b><a NAME="E9:79_1_1"/>
1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_000.html#K18">.size()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/></div><b>end;</b></div><a NAME="E4:79_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:79_1_2"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being <a href="glib_000.html#V2">finite</a> <a href="glib_002.html#V3" target="_self">Tree-like</a> <a href="glib_000.html#NM2">_Graph</a> st <font color="Olive">b<sub>1</sub></font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="glib_000.html#K18">.size()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="glib_000.html#V2">finite</a> <a href="glib_002.html#V3" target="_self">Tree-like</a> <a href="glib_000.html#NM2">_Graph</a>;<br/><b>assume </b><a NAME="E2:79_1_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E3:79_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T3">XCMPLX_1:3</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as  <a href="glib_000.html#V2">finite</a> non <a href="glib_000.html#V4">trivial</a> <a href="glib_002.html#V3" target="_self">Tree-like</a> <a href="glib_000.html#NM2">_Graph</a> <b>by </b><i><a class="ref" href="glib_000.html#T29">GLIB_000:29</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>5</sub></font> being  <a href="glib_000.html#V11">endvertex</a> <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>4</sub></font>;<br/><b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="glib_000.html#NM6">removeVertex</a> of <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>;<br/><a NAME="E7:79_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="txt" href="glib_002.html#E2:79_1_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="glib_000.html#T51">GLIB_000:51</a></i>;<br/><a NAME="E8:79_1_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 ;<br/><a NAME="E9:79_1_2"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_000.html#K18">.size()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="glib_002.html#E1:79_1_2"><i><font color="Green">E50</font></i></a></i>;<br/> <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="glib_000.html#K29">.edgesInOut()</a> </span>)</span> = 
<font color="Maroon">c<sub>5</sub></font> <a href="glib_000.html#K36">.degree()</a> 
<b>by </b><i><a class="ref" href="glib_000.html#T22">GLIB_000:22</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="glib_000.html#D54">GLIB_000:def 54</a></i>
;<br/><b>hence </b><a NAME="E11:79_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="glib_000.html#K18">.size()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="glib_002.html#E9:79_1_2"><i><font color="Green">E52</font></i></a>, <a class="ref" href="glib_000.html#T51">GLIB_000:51</a></i>;<br/></div><b>end;</b></div><a NAME="E6:79_1"/><b>then </b><i><font color="Green">E50</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/><a NAME="E7:79_1"/>
for <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="binarith.html#S1">BINARITH:sch 1</a>(<a class="txt" href="glib_002.html#E4:79_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="glib_002.html#E6:79_1"><i><font color="Green">E50</font></i></a>);<br/></i><b>hence </b><a NAME="E8:79_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K18">.size()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="glib_002.html#E1:79_1"><i><font color="Green">E48</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:79"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="glib_002.html#V2" target="_self">acyclic</a> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K18">.size()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means for <font color="Olive">b<sub>1</sub></font> being <a href="glib_000.html#V2">finite</a> <a href="glib_002.html#V2" target="_self">acyclic</a> <a href="glib_000.html#NM2">_Graph</a> st <font color="Olive">b<sub>1</sub></font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="glib_000.html#K18">.size()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1 holds <br/><font color="Olive">b<sub>1</sub></font> is <a href="glib_002.html#V1" target="_self">connected</a>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="glib_000.html#V2">finite</a> <a href="glib_002.html#V2" target="_self">acyclic</a> <a href="glib_000.html#NM2">_Graph</a>;<br/><b>assume </b><a NAME="E1:79_2"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_000.html#K18">.size()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><b>then consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:79_2"/><i><font color="Green">E49</font></i>: 
 <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="glib_000.html#T30">GLIB_000:30</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E1:79_2_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="glib_002.html#E3:79_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E2:79_2_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K1">.walkOf</a> <font color="Maroon">c<sub>3</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="glib_001.html#T14">GLIB_001:14</a></i>;<br/><b>hence </b><a NAME="E3:79_2_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:79_2"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="glib_002.html#V1" target="_self">connected</a>
 <b>by </b><i><a class="ref" href="glib_002.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:79"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:79_3"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being <a href="glib_000.html#V2">finite</a> <a href="glib_002.html#V2" target="_self">acyclic</a> <a href="glib_000.html#NM2">_Graph</a> st <font color="Olive">b<sub>1</sub></font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="glib_000.html#K18">.size()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1 holds <br/><font color="Olive">b<sub>1</sub></font> is <a href="glib_002.html#V1" target="_self">connected</a>
 ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="glib_000.html#V2">finite</a> <a href="glib_002.html#V2" target="_self">acyclic</a> <a href="glib_000.html#NM2">_Graph</a>;<br/><b>assume </b><a NAME="E2:79_3"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="glib_000.html#K18">.size()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:79_3_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> 1
 ;<br/><a NAME="E2:79_3_1"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="glib_002.html#E2:79_3"><i><font color="Green">E51</font></i></a></i>;<br/><b>hence </b><a NAME="E3:79_3_1"/>
contradiction
 ;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as  <a href="glib_000.html#V2">finite</a> non <a href="glib_000.html#V4">trivial</a> <a href="glib_002.html#V2" target="_self">acyclic</a> <a href="glib_000.html#NM2">_Graph</a> <b>by </b><i><a class="ref" href="glib_000.html#T29">GLIB_000:29</a></i>;<br/><a NAME="E5:79_3"/>
 <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="glib_002.html#E2:79_3"><i><font color="Green">E51</font></i></a>, <a class="ref" href="card_1.html#T78">CARD_1:78</a>, <a class="ref" href="glib_000.html#D27">GLIB_000:def 27</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E7:79_3"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="glib_000.html#V11">endvertex</a> &amp; <font color="Maroon">c<sub>6</sub></font> is <a href="glib_000.html#V11">endvertex</a> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="glib_002.html#K1" target="_self">.reachableFrom</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="glib_002.html#E36"><i><font color="Green">Lemma35</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="glib_000.html#NM6">removeVertex</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>;<br/><a NAME="E9:79_3"/><i><font color="Green">E53</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="glib_000.html#K16">.order()</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="glib_000.html#K18">.size()</a> </span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="glib_000.html#K29">.edgesInOut()</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="glib_000.html#K18">.size()</a>  )
 <b>by </b><i><a class="ref" href="glib_000.html#T51">GLIB_000:51</a></i>;<br/><a NAME="E10:79_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="glib_002.html#E2:79_3"><i><font color="Green">E51</font></i></a>, <a class="ref" href="glib_000.html#T51">GLIB_000:51</a></i>;<br/><a NAME="E11:79_3"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/> <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="glib_000.html#K29">.edgesInOut()</a> </span>)</span> = 
<font color="Maroon">c<sub>5</sub></font> <a href="glib_000.html#K36">.degree()</a> 
<b>by </b><i><a class="ref" href="glib_000.html#T22">GLIB_000:22</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="glib_002.html#E7:79_3"><i><font color="Green">E52</font></i></a>, <a class="ref" href="glib_000.html#D54">GLIB_000:def 54</a></i>
;<br/><a NAME="E13:79_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="glib_000.html#K18">.size()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="glib_002.html#E2:79_3"><i><font color="Green">E51</font></i></a>, <a class="txt" href="glib_002.html#E9:79_3"><i><font color="Green">E53</font></i></a></i>;<br/><a NAME="E14:79_3"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>7</sub></font> is <a href="glib_002.html#V1" target="_self">connected</a>
 <b>by </b><i><a class="txt" href="glib_002.html#E1:79_3"><i><font color="Green">E50</font></i></a>, <a class="txt" href="glib_002.html#E11:79_3"><i><font color="Green">E54</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E16:79_3"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="glib_000.html#K29">.edgesInOut()</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K1">}</a></span> &amp; not <font color="Maroon">c<sub>8</sub></font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="glib_002.html#E7:79_3"><i><font color="Green">E52</font></i></a>, <a class="ref" href="glib_000.html#D53">GLIB_000:def 53</a></i>;<br/><a NAME="E17:79_3"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="glib_000.html#K29">.edgesInOut()</a> 
 <b>by </b><i><a class="txt" href="glib_002.html#E16:79_3"><i><font color="Green">E56</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E18:79_3"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="glib_002.html#V1" target="_self">connected</a>
 <b>by </b><i><a class="txt" href="glib_002.html#E14:79_3"><i><font color="Green">E55</font></i></a>, <a class="txt" href="glib_002.html#E16:79_3"><i><font color="Green">E56</font></i></a>, <a class="ref" href="glib_002.html#T3" target="_self">Th3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:79"/><b>then </b><i><font color="Green">E50</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
;<br/>
<a NAME="E7:79"/>
for <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="binarith.html#S1">BINARITH:sch 1</a>(<a class="txt" href="glib_002.html#E4:79"><i><font color="Green">E49</font></i></a>, <a class="txt" href="glib_002.html#E6:79"><i><font color="Green">E50</font></i></a>);<br/></i>
<a NAME="E8:79"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> is <a href="glib_002.html#V1" target="_self">connected</a>
 
<b>by </b><i><a class="txt" href="glib_002.html#E2:79"><i><font color="Green">E48</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:79"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="glib_002.html#V3" target="_self">Tree-like</a>
 <b>by </b><i><a class="txt" href="glib_002.html#E2:79"><i><font color="Green">E48</font></i></a>, <a class="ref" href="glib_002.html#D3" target="_self">Def3</a></i>;<br/>


</div>
