<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">G</font> be   <a href="glib_000.html#NM2">_Graph</a>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:46_1"/>
<font color="Maroon">G</font> is <a href="glib_002.html#V1" target="_self">connected</a>
 ;<br/><a NAME="E2:46_1"/><b>then </b>
<font color="Maroon">G</font> <a href="glib_002.html#K3" target="_self">.componentSet()</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i/>;<br/><a NAME="E3:46_1"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_002.html#K3" target="_self">.componentSet()</a> </span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="card_1.html#T50">CARD_1:50</a></i>;<br/><b>hence </b><a NAME="E4:46_1"/>
<font color="Maroon">G</font> <a href="glib_002.html#K4" target="_self">.numComponents()</a>  <a href="hidden.html#R1">=</a> 1
 ;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:46"/>
<font color="Maroon">G</font> <a href="glib_002.html#K4" target="_self">.numComponents()</a>  <a href="hidden.html#R1">=</a> 1
 ;<br/>

<a NAME="E3:46"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_002.html#K3" target="_self">.componentSet()</a> </span>)</span> <a href="hidden.html#R1">=</a> 1
 
;<br/>
<b>then consider </b><font color="Maroon">V</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:46"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">G</font> <a href="glib_002.html#K3" target="_self">.componentSet()</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">V</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/>
<a NAME="E6:46"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="glib_002.html#K3" target="_self">.componentSet()</a> 
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">v</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:46_2"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font>
 ;<br/><b>then reconsider </b><font color="Maroon">v'</font> = <font color="Maroon">v</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:46_2_1"/><i><font color="Green">E43</font></i>: 
not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 ;<br/><b>set </b><font color="Maroon">V2</font> = <font color="Maroon">G</font> <a href="glib_002.html#K1" target="_self">.reachableFrom</a> <font color="Maroon">v'</font>;<br/><a NAME="E2:46_2_1"/>
<font color="Maroon">G</font> <a href="glib_002.html#K1" target="_self">.reachableFrom</a> <font color="Maroon">v'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="glib_002.html#K3" target="_self">.componentSet()</a> 
 <b>by </b><i/>;<br/><a NAME="E3:46_2_1"/><b>then </b>
not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="glib_002.html#K1" target="_self">.reachableFrom</a> <font color="Maroon">v'</font>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:46_2_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:46_2"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E8:46"/><b>then </b>
 <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E9:46"/><b>then </b>
<font color="Maroon">V</font> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E10:46"/>
<font color="Maroon">G</font> is <a href="glib_002.html#V1" target="_self">connected</a>
 <b>by </b><i>, </i>;<br/>


</div>
