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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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> holds <br/> ( not <font color="Olive" title="b1">v</font> is  <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a>  iff 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> holds  <font color="Olive" title="b2">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> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K4" title="GLIB_002:func.4">.numComponents()</a>  )</span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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>  iff 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="Olive" title="b1">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> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K4" title="GLIB_002:func.4">.numComponents()</a>  )</span><br/>



<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="Olive" title="b1">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> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K4" title="GLIB_002:func.4">.numComponents()</a>  ) implies  not <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"><span class="kw">assume </span><a NAME="E1:46_1"/>
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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="Olive" title="b1">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> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K4" title="GLIB_002:func.4">.numComponents()</a> </span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">G3</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><span class="kw"> such that </span><br/><a NAME="E3:46_1"/><span class="lab"><font color="Green" title="E31">A1</font></span>: 
not <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K4" title="GLIB_002:func.4">.numComponents()</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">G3</font> <a href="glib_002.html#K4" title="GLIB_002:func.4">.numComponents()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_002.html#D10" target="_self" title="GLIB_002:def.10">Def10</a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c4">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">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> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K4" title="GLIB_002:func.4">.numComponents()</a> </span><br/><a NAME="E4:46_1"/>
<font color="Maroon" title="c3">G3</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> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K4" title="GLIB_002:func.4">.numComponents()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46_1"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="ref" href="card_1.html#T14" title="CARD_1:th.14">CARD_1:14</a></span>;<br/><span class="kw">hence </span><a NAME="E5:46_1"/>
<font color="Maroon" title="c4">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> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K4" title="GLIB_002:func.4">.numComponents()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="glib_002.html#E24"><span class="lab"><font color="Green" title="E23">Lm12</font></span></a>, <a class="ref" href="glib_000.html#T96" title="GLIB_000:th.96">GLIB_000:96</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:46"/><span class="lab"><font color="Green" title="E31">A2</font></span>: 
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="Olive" title="b1">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> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K4" title="GLIB_002:func.4">.numComponents()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_2"><span class="kw">now </span></a><div class="add"><span class="kw">consider </span><font color="Maroon" title="c3">X</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>;<br/><span class="kw">assume </span><a NAME="E2:46_2"/>
for <font color="Olive" title="b1">G3</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#K4" title="GLIB_002:func.4">.numComponents()</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">G3</font> <a href="glib_002.html#K4" title="GLIB_002:func.4">.numComponents()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E3:46_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A3</font></span>: 
<font color="Maroon" title="c1">G</font> <a href="glib_002.html#K4" title="GLIB_002:func.4">.numComponents()</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="glib_002.html#K4" title="GLIB_002:func.4">.numComponents()</a> 
 ;<br/><a NAME="E4:46_2"/>
<font color="Maroon" title="c3">X</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> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K4" title="GLIB_002:func.4">.numComponents()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46"><span class="lab"><font color="Green" title="E31">A2</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:46_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46_2"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="ref" href="card_1.html#T14" title="CARD_1:th.14">CARD_1:14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:46"/>
not <font color="Maroon" title="c2">v</font> is  <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_002.html#D10" target="_self" title="GLIB_002:def.10">Def10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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