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<b>let </b><font color="Maroon" title="c1">G</font> be  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">v</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font><br/> for <font color="Olive" title="b2">G2</font> being  <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">G</font>,<font color="Olive" title="b1">v</font>  st not <font color="Olive" title="b1">v</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> holds <br/><font color="Olive" title="b2">G2</font> is <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a></span><br/><b>let </b><font color="Maroon" title="c2">v</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">G2</font> being  <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font>  st not <font color="Maroon" title="c2">v</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> holds <br/><font color="Olive" title="b1">G2</font> is <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a></span><br/><b>let </b><font color="Maroon" title="c3">G2</font> be   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not <font color="Maroon" title="c2">v</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> implies <font color="Maroon" title="c3">G2</font> is <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> )</span><br/>





<b>assume </b><a NAME="E1:47"/><i><font color="Green" title="E32">A1</font></i>: 
not <font color="Maroon" title="c2">v</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">G2</font> is <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a></span><br/>

<a NAME="E2:47"/>
<font color="Maroon" title="c1">G</font> <a href="glib_002.html#K4" title="GLIB_002:func.4">.numComponents()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="glib_002.html#E30"><i><font color="Green" title="E29">Lm18</font></i></a></i>;<br/>
<a NAME="E3:47"/><b>then </b>
<font color="Maroon" title="c3">G2</font> <a href="glib_002.html#K4" title="GLIB_002:func.4">.numComponents()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="glib_002.html#E1:47"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="glib_002.html#E31"><i><font color="Green" title="E30">Lm19</font></i></a></i>;<br/>
<a NAME="E4:47"/><b>then </b>
<font color="Maroon" title="c3">G2</font> <a href="glib_002.html#K4" title="GLIB_002:func.4">.numComponents()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="ordinal1.html#D1" title="ORDINAL1:def.1">ORDINAL1:def 1</a></i>;<br/>
<a NAME="E5:47"/><b>then </b><i><font color="Green" title="E33">A2</font></i>: 
( <font color="Maroon" title="c3">G2</font> <a href="glib_002.html#K4" title="GLIB_002:func.4">.numComponents()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c3">G2</font> <a href="glib_002.html#K4" title="GLIB_002:func.4">.numComponents()</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"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T39" title="ZFMISC_1:th.39">ZFMISC_1:39</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">x</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c3">G2</font>;<br/>
<font color="Maroon" title="c3">G2</font> <a href="glib_002.html#K4" title="GLIB_002:func.4">.numComponents()</a>  = 
<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="txt" href="glib_002.html#E5:47"><i><font color="Green" title="E33">A2</font></i></a></i>
<br/>.= 
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="ordinal1.html#D1" title="ORDINAL1:def.1">ORDINAL1:def 1</a></i>
<br/>.= 
1

;<br/>
<b>hence </b><a NAME="E8:47"/>
<font color="Maroon" title="c3">G2</font> is <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>
 <b>by </b><i><a class="txt" href="glib_002.html#E30"><i><font color="Green" title="E29">Lm18</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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