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<b>let </b><font color="Maroon">G</font> be  <a href="glib_002.html#V1" target="_self">connected</a> <a href="glib_000.html#NM2">_Graph</a>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">G2</font> be   <a href="glib_000.html#NM6">removeVertex</a> of <font color="Maroon">G</font>,<font color="Maroon">v</font>;<br/>





<b>assume </b><a NAME="E1:48"/><i><font color="Green">E41</font></i>: 
not <font color="Maroon">v</font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a>
 ;<br/>

<a NAME="E2:48"/>
<font color="Maroon">G</font> <a href="glib_002.html#K4" target="_self">.numComponents()</a>  <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E3:48"/><b>then </b>
<font color="Maroon">G2</font> <a href="glib_002.html#K4" target="_self">.numComponents()</a>  <a href="ordinal1.html#R1">c=</a>  <a href="ordinal1.html#K1">succ</a> 0
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:48"/><b>then </b>
<font color="Maroon">G2</font> <a href="glib_002.html#K4" target="_self">.numComponents()</a>  <a href="tarski.html#R1">c=</a> <a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="ordinal1.html#D1">ORDINAL1:def 1</a></i>;<br/>
<a NAME="E5:48"/><b>then </b><i><font color="Green">E42</font></i>: 
( <font color="Maroon">G2</font> <a href="glib_002.html#K4" target="_self">.numComponents()</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">G2</font> <a href="glib_002.html#K4" target="_self">.numComponents()</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T39">ZFMISC_1:39</a></i>;<br/>
<b>consider </b><font color="Maroon">x</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G2</font>;<br/>
<a NAME="E7:48"/>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="glib_002.html#K3" target="_self">.componentSet()</a> </span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="card_2.html#T59">CARD_2:59</a></i>;<br/>
<b>then </b><font color="Maroon">G2</font> <a href="glib_002.html#K4" target="_self">.numComponents()</a>  = 
<a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
<b>by </b><i/>
<br/>.= 
 <a href="ordinal1.html#K1">succ</a> 0
<b>by </b><i><a class="ref" href="ordinal1.html#D1">ORDINAL1:def 1</a></i>
<br/>.= 
1

;<br/>
<b>hence </b><a NAME="E9:48"/>
<font color="Maroon">G2</font> is <a href="glib_002.html#V1" target="_self">connected</a>
 <b>by </b><i/>;<br/>


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