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<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:32_1_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">v</font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a>
 ;<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/><a NAME="E2:32_1_1_1"/>
<font color="Maroon">G</font> <a href="glib_002.html#K5" target="_self">.numComponents()</a>  <a href="card_1.html#NR2" target="_self">&lt;`</a> <font color="Maroon">G2</font> <a href="glib_002.html#K5" target="_self">.numComponents()</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E3:32_1_1_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#K5" target="_self">.numComponents()</a> </span>)</span> <a href="card_1.html#NR2" target="_self">&lt;`</a> <font color="Maroon">G2</font> <a href="glib_002.html#K5" target="_self">.numComponents()</a> 
 <b>by </b><i><a class="ref" href="card_1.html#T66">CARD_1:66</a></i>;<br/><a NAME="E4:32_1_1_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#K5" target="_self">.numComponents()</a> </span>)</span> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="glib_002.html#K5" target="_self">.numComponents()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="card_1.html#T66">CARD_1:66</a></i>;<br/><b>hence </b><a NAME="E5:32_1_1_1"/>
<font color="Maroon">G</font> <a href="glib_002.html#K5" target="_self">.numComponents()</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">G2</font> <a href="glib_002.html#K5" target="_self">.numComponents()</a> 
 <b>by </b><i><a class="ref" href="card_1.html#T73">CARD_1:73</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:32_1_1"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">G2</font> being  <a href="glib_000.html#NM6">removeVertex</a> of <font color="Maroon">G</font>,<font color="Maroon">v</font>  holds <font color="Maroon">G</font> <a href="glib_002.html#K5" target="_self">.numComponents()</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">G2</font> <a href="glib_002.html#K5" target="_self">.numComponents()</a> 
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_2"><b>now </b></a><div class="add"><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/><a NAME="E1:32_1_1_2"/>
<font color="Maroon">G</font> <a href="glib_002.html#K5" target="_self">.numComponents()</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">G2</font> <a href="glib_002.html#K5" target="_self">.numComponents()</a> 
 <b>by </b><i/>;<br/><a NAME="E2:32_1_1_2"/><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#K5" target="_self">.numComponents()</a> </span>)</span> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="glib_002.html#K5" target="_self">.numComponents()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="card_1.html#T73">CARD_1:73</a></i>;<br/><a NAME="E3:32_1_1_2"/><b>then </b>
<font color="Maroon">G</font> <a href="glib_002.html#K5" target="_self">.numComponents()</a>  <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="glib_002.html#K5" target="_self">.numComponents()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="card_1.html#T66">CARD_1:66</a></i>;<br/><b>hence </b><a NAME="E4:32_1_1_2"/>
<font color="Maroon">G</font> <a href="glib_002.html#K5" target="_self">.numComponents()</a>  <a href="card_1.html#NR2" target="_self">&lt;`</a> <font color="Maroon">G2</font> <a href="glib_002.html#K5" target="_self">.numComponents()</a> 
 <b>by </b><i><a class="ref" href="card_1.html#T66">CARD_1:66</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:32_1_1"/>
<font color="Maroon">v</font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a>
 <b>by </b><i/>;<br/>


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