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

<div><b>hereby </b> <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>  holds <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K5" title="GLIB_002:func.5">.numComponents()</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">G2</font> <a href="glib_002.html#K5" title="GLIB_002:func.5">.numComponents()</a>  ) implies <font color="Maroon" title="c2">v</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> )</span>
<div class="add"><b>assume </b><a NAME="E1:31_1_1_1"/><i><font color="Green" title="E15">A1</font></i>: 
<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"> 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>  holds <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K5" title="GLIB_002:func.5">.numComponents()</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">G2</font> <a href="glib_002.html#K5" title="GLIB_002:func.5">.numComponents()</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"> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K5" title="GLIB_002:func.5">.numComponents()</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">G2</font> <a href="glib_002.html#K5" title="GLIB_002:func.5">.numComponents()</a> </span><br/><a NAME="E2:31_1_1_1"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K5" title="GLIB_002:func.5">.numComponents()</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G2</font> <a href="glib_002.html#K5" title="GLIB_002:func.5">.numComponents()</a> </span>)</span>
 <b>by </b><i><a class="txt" href="#E1:31_1_1_1"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="glib_002.html#D10" target="_self" title="GLIB_002:def.10">Def10</a></i>;<br/><b>hence </b><a NAME="E3:31_1_1_1"/>
<font color="Maroon" title="c1">G</font> <a href="glib_002.html#K5" title="GLIB_002:func.5">.numComponents()</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">G2</font> <a href="glib_002.html#K5" title="GLIB_002:func.5">.numComponents()</a> 
 <b>by </b><i><a class="ref" href="nat_1.html#T42" title="NAT_1:th.42">NAT_1:42</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:31_1_1"/><i><font color="Green" title="E15">A2</font></i>: 
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>  holds <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K5" title="GLIB_002:func.5">.numComponents()</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">G2</font> <a href="glib_002.html#K5" title="GLIB_002:func.5">.numComponents()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">v</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_2"><b>now </b></a><div class="add"><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"> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K5" title="GLIB_002:func.5">.numComponents()</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">G2</font> <a href="glib_002.html#K5" title="GLIB_002:func.5">.numComponents()</a> </span><br/><a NAME="E1:31_1_1_2"/>
<font color="Maroon" title="c1">G</font> <a href="glib_002.html#K5" title="GLIB_002:func.5">.numComponents()</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">G2</font> <a href="glib_002.html#K5" title="GLIB_002:func.5">.numComponents()</a> 
 <b>by </b><i><a class="txt" href="#E2:31_1_1"><i><font color="Green" title="E15">A2</font></i></a></i>;<br/><a NAME="E2:31_1_1_2"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K5" title="GLIB_002:func.5">.numComponents()</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G2</font> <a href="glib_002.html#K5" title="GLIB_002:func.5">.numComponents()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T42" title="NAT_1:th.42">NAT_1:42</a></i>;<br/><b>hence </b><a NAME="E3:31_1_1_2"/>
<font color="Maroon" title="c1">G</font> <a href="glib_002.html#K5" title="GLIB_002:func.5">.numComponents()</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">G2</font> <a href="glib_002.html#K5" title="GLIB_002:func.5">.numComponents()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:31_1_1"/>
<font color="Maroon" title="c2">v</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a>
 <b>by </b><i><a class="ref" href="glib_002.html#D10" target="_self" title="GLIB_002:def.10">Def10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
